Properties

 Degree 88 Conductor $2^{132} \cdot 751^{44}$ Sign $1$ Motivic weight 1 Primitive no Self-dual yes Analytic rank 44

Origins of factors

Dirichlet series

 L(s)  = 1 − 14·3-s + 7·5-s − 20·7-s + 51·9-s − 19·11-s − 10·13-s − 98·15-s − 16·17-s − 25·19-s + 280·21-s − 29·23-s − 71·25-s + 198·27-s + 35·29-s − 49·31-s + 266·33-s − 140·35-s − 30·37-s + 140·39-s − 14·41-s − 35·43-s + 357·45-s − 45·47-s + 56·49-s + 224·51-s + 18·53-s − 133·55-s + ⋯
 L(s)  = 1 − 8.08·3-s + 3.13·5-s − 7.55·7-s + 17·9-s − 5.72·11-s − 2.77·13-s − 25.3·15-s − 3.88·17-s − 5.73·19-s + 61.1·21-s − 6.04·23-s − 14.1·25-s + 38.1·27-s + 6.49·29-s − 8.80·31-s + 46.3·33-s − 23.6·35-s − 4.93·37-s + 22.4·39-s − 2.18·41-s − 5.33·43-s + 53.2·45-s − 6.56·47-s + 8·49-s + 31.3·51-s + 2.47·53-s − 17.9·55-s + ⋯

Functional equation

\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{132} \cdot 751^{44}\right)^{s/2} \, \Gamma_{\C}(s)^{44} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{132} \cdot 751^{44}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{44} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}

Invariants

 $$d$$ = $$88$$ $$N$$ = $$2^{132} \cdot 751^{44}$$ $$\varepsilon$$ = $1$ motivic weight = $$1$$ character : induced by $\chi_{6008} (1, \cdot )$ primitive : no self-dual : yes analytic rank = 44 Selberg data = $(88,\ 2^{132} \cdot 751^{44} ,\ ( \ : [1/2]^{44} ),\ 1 )$ $L(1)$ $=$ $0$ $L(\frac12)$ $=$ $0$ $L(\frac{3}{2})$ not available $L(1)$ not available

Euler product

$L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1}$where, for $p \notin \{2,\;751\}$,$$F_p(T)$$ is a polynomial of degree 88. If $p \in \{2,\;751\}$, then $F_p(T)$ is a polynomial of degree at most 87.
$p$$F_p(T)$
bad2 $$1$$
751 $$( 1 + T )^{44}$$
good3 $$1 + 14 T + 145 T^{2} + 1118 T^{3} + 7370 T^{4} + 42128 T^{5} + 217330 T^{6} + 340820 p T^{7} + 4463350 T^{8} + 18210106 T^{9} + 288427 p^{5} T^{10} + 28420741 p^{2} T^{11} + 296751931 p T^{12} + 2966049187 T^{13} + 9496354592 T^{14} + 29300327597 T^{15} + 87365822663 T^{16} + 252293772094 T^{17} + 707103587228 T^{18} + 642231246425 p T^{19} + 5112176884702 T^{20} + 13226811131449 T^{21} + 11137864081177 p T^{22} + 82506553674592 T^{23} + 199343772770006 T^{24} + 471688018449241 T^{25} + 1093982864268991 T^{26} + 2488782072152794 T^{27} + 5557517447240399 T^{28} + 12188518317302960 T^{29} + 26268677309346641 T^{30} + 55661192063624291 T^{31} + 116008178638122040 T^{32} + 79303799020093925 p T^{33} + 160090990074678542 p T^{34} + 954642617570774044 T^{35} + 1868940109841669408 T^{36} + 400508831036949070 p^{2} T^{37} + 6850323702587989324 T^{38} + 12830399360324567089 T^{39} + 877292037757387132 p^{3} T^{40} + 14369558668659811241 p T^{41} + 77348527502218161551 T^{42} + 15203874683987507030 p^{2} T^{43} +$$$$23\!\cdots\!40$$$$T^{44} + 15203874683987507030 p^{3} T^{45} + 77348527502218161551 p^{2} T^{46} + 14369558668659811241 p^{4} T^{47} + 877292037757387132 p^{7} T^{48} + 12830399360324567089 p^{5} T^{49} + 6850323702587989324 p^{6} T^{50} + 400508831036949070 p^{9} T^{51} + 1868940109841669408 p^{8} T^{52} + 954642617570774044 p^{9} T^{53} + 160090990074678542 p^{11} T^{54} + 79303799020093925 p^{12} T^{55} + 116008178638122040 p^{12} T^{56} + 55661192063624291 p^{13} T^{57} + 26268677309346641 p^{14} T^{58} + 12188518317302960 p^{15} T^{59} + 5557517447240399 p^{16} T^{60} + 2488782072152794 p^{17} T^{61} + 1093982864268991 p^{18} T^{62} + 471688018449241 p^{19} T^{63} + 199343772770006 p^{20} T^{64} + 82506553674592 p^{21} T^{65} + 11137864081177 p^{23} T^{66} + 13226811131449 p^{23} T^{67} + 5112176884702 p^{24} T^{68} + 642231246425 p^{26} T^{69} + 707103587228 p^{26} T^{70} + 252293772094 p^{27} T^{71} + 87365822663 p^{28} T^{72} + 29300327597 p^{29} T^{73} + 9496354592 p^{30} T^{74} + 2966049187 p^{31} T^{75} + 296751931 p^{33} T^{76} + 28420741 p^{35} T^{77} + 288427 p^{39} T^{78} + 18210106 p^{35} T^{79} + 4463350 p^{36} T^{80} + 340820 p^{38} T^{81} + 217330 p^{38} T^{82} + 42128 p^{39} T^{83} + 7370 p^{40} T^{84} + 1118 p^{41} T^{85} + 145 p^{42} T^{86} + 14 p^{43} T^{87} + p^{44} T^{88}$$
5 $$1 - 7 T + 24 p T^{2} - 147 p T^{3} + 7102 T^{4} - 38899 T^{5} + 55572 p T^{6} - 276616 p T^{7} + 8111126 T^{8} - 37150409 T^{9} + 188827583 T^{10} - 803670001 T^{11} + 3656560894 T^{12} - 583071343 p^{2} T^{13} + 60630500657 T^{14} - 227869346124 T^{15} + 879198475066 T^{16} - 3132142411591 T^{17} + 11329586638171 T^{18} - 38433427721014 T^{19} + 26275099175061 p T^{20} - 426022322476266 T^{21} + 1384685282342411 T^{22} - 4306580044082367 T^{23} + 2674982169217166 p T^{24} - 8002155114652109 p T^{25} + 119204917723397684 T^{26} - 550147090946989 p^{4} T^{27} + 985930193195216044 T^{28} - 549604306059831649 p T^{29} + 7604113476815615849 T^{30} - 4103614170835693647 p T^{31} + 54913993881537264229 T^{32} - 28734601645338968842 p T^{33} +$$$$37\!\cdots\!59$$$$T^{34} -$$$$94\!\cdots\!23$$$$T^{35} +$$$$23\!\cdots\!97$$$$T^{36} -$$$$58\!\cdots\!53$$$$T^{37} +$$$$14\!\cdots\!71$$$$T^{38} -$$$$69\!\cdots\!36$$$$p T^{39} +$$$$82\!\cdots\!02$$$$T^{40} -$$$$38\!\cdots\!54$$$$p T^{41} +$$$$44\!\cdots\!71$$$$T^{42} -$$$$10\!\cdots\!06$$$$T^{43} +$$$$22\!\cdots\!28$$$$T^{44} -$$$$10\!\cdots\!06$$$$p T^{45} +$$$$44\!\cdots\!71$$$$p^{2} T^{46} -$$$$38\!\cdots\!54$$$$p^{4} T^{47} +$$$$82\!\cdots\!02$$$$p^{4} T^{48} -$$$$69\!\cdots\!36$$$$p^{6} T^{49} +$$$$14\!\cdots\!71$$$$p^{6} T^{50} -$$$$58\!\cdots\!53$$$$p^{7} T^{51} +$$$$23\!\cdots\!97$$$$p^{8} T^{52} -$$$$94\!\cdots\!23$$$$p^{9} T^{53} +$$$$37\!\cdots\!59$$$$p^{10} T^{54} - 28734601645338968842 p^{12} T^{55} + 54913993881537264229 p^{12} T^{56} - 4103614170835693647 p^{14} T^{57} + 7604113476815615849 p^{14} T^{58} - 549604306059831649 p^{16} T^{59} + 985930193195216044 p^{16} T^{60} - 550147090946989 p^{21} T^{61} + 119204917723397684 p^{18} T^{62} - 8002155114652109 p^{20} T^{63} + 2674982169217166 p^{21} T^{64} - 4306580044082367 p^{21} T^{65} + 1384685282342411 p^{22} T^{66} - 426022322476266 p^{23} T^{67} + 26275099175061 p^{25} T^{68} - 38433427721014 p^{25} T^{69} + 11329586638171 p^{26} T^{70} - 3132142411591 p^{27} T^{71} + 879198475066 p^{28} T^{72} - 227869346124 p^{29} T^{73} + 60630500657 p^{30} T^{74} - 583071343 p^{33} T^{75} + 3656560894 p^{32} T^{76} - 803670001 p^{33} T^{77} + 188827583 p^{34} T^{78} - 37150409 p^{35} T^{79} + 8111126 p^{36} T^{80} - 276616 p^{38} T^{81} + 55572 p^{39} T^{82} - 38899 p^{39} T^{83} + 7102 p^{40} T^{84} - 147 p^{42} T^{85} + 24 p^{43} T^{86} - 7 p^{43} T^{87} + p^{44} T^{88}$$
7 $$1 + 20 T + 344 T^{2} + 4216 T^{3} + 45981 T^{4} + 428424 T^{5} + 522251 p T^{6} + 28254854 T^{7} + 203811974 T^{8} + 1368391963 T^{9} + 8684709636 T^{10} + 52100506575 T^{11} + 298124949797 T^{12} + 1628447179040 T^{13} + 8540475771857 T^{14} + 43049682839025 T^{15} + 209392597942006 T^{16} + 983785624968252 T^{17} + 4477485232277049 T^{18} + 2822746154453479 p T^{19} + 84731948712193255 T^{20} + 50481117324896063 p T^{21} + 1435622769817671123 T^{22} + 5685924821167155296 T^{23} + 3140465616794198033 p T^{24} + 11860012427255683820 p T^{25} + 43797496624871320644 p T^{26} +$$$$11\!\cdots\!46$$$$T^{27} +$$$$55\!\cdots\!87$$$$p T^{28} +$$$$13\!\cdots\!35$$$$T^{29} +$$$$46\!\cdots\!84$$$$T^{30} +$$$$15\!\cdots\!55$$$$T^{31} +$$$$50\!\cdots\!90$$$$T^{32} +$$$$16\!\cdots\!95$$$$T^{33} +$$$$50\!\cdots\!61$$$$T^{34} +$$$$15\!\cdots\!88$$$$T^{35} +$$$$48\!\cdots\!32$$$$T^{36} +$$$$14\!\cdots\!28$$$$T^{37} +$$$$42\!\cdots\!42$$$$T^{38} +$$$$12\!\cdots\!93$$$$T^{39} +$$$$71\!\cdots\!34$$$$p^{2} T^{40} +$$$$97\!\cdots\!61$$$$T^{41} +$$$$26\!\cdots\!27$$$$T^{42} +$$$$10\!\cdots\!62$$$$p T^{43} +$$$$19\!\cdots\!52$$$$T^{44} +$$$$10\!\cdots\!62$$$$p^{2} T^{45} +$$$$26\!\cdots\!27$$$$p^{2} T^{46} +$$$$97\!\cdots\!61$$$$p^{3} T^{47} +$$$$71\!\cdots\!34$$$$p^{6} T^{48} +$$$$12\!\cdots\!93$$$$p^{5} T^{49} +$$$$42\!\cdots\!42$$$$p^{6} T^{50} +$$$$14\!\cdots\!28$$$$p^{7} T^{51} +$$$$48\!\cdots\!32$$$$p^{8} T^{52} +$$$$15\!\cdots\!88$$$$p^{9} T^{53} +$$$$50\!\cdots\!61$$$$p^{10} T^{54} +$$$$16\!\cdots\!95$$$$p^{11} T^{55} +$$$$50\!\cdots\!90$$$$p^{12} T^{56} +$$$$15\!\cdots\!55$$$$p^{13} T^{57} +$$$$46\!\cdots\!84$$$$p^{14} T^{58} +$$$$13\!\cdots\!35$$$$p^{15} T^{59} +$$$$55\!\cdots\!87$$$$p^{17} T^{60} +$$$$11\!\cdots\!46$$$$p^{17} T^{61} + 43797496624871320644 p^{19} T^{62} + 11860012427255683820 p^{20} T^{63} + 3140465616794198033 p^{21} T^{64} + 5685924821167155296 p^{21} T^{65} + 1435622769817671123 p^{22} T^{66} + 50481117324896063 p^{24} T^{67} + 84731948712193255 p^{24} T^{68} + 2822746154453479 p^{26} T^{69} + 4477485232277049 p^{26} T^{70} + 983785624968252 p^{27} T^{71} + 209392597942006 p^{28} T^{72} + 43049682839025 p^{29} T^{73} + 8540475771857 p^{30} T^{74} + 1628447179040 p^{31} T^{75} + 298124949797 p^{32} T^{76} + 52100506575 p^{33} T^{77} + 8684709636 p^{34} T^{78} + 1368391963 p^{35} T^{79} + 203811974 p^{36} T^{80} + 28254854 p^{37} T^{81} + 522251 p^{39} T^{82} + 428424 p^{39} T^{83} + 45981 p^{40} T^{84} + 4216 p^{41} T^{85} + 344 p^{42} T^{86} + 20 p^{43} T^{87} + p^{44} T^{88}$$
11 $$1 + 19 T + 411 T^{2} + 494 p T^{3} + 71734 T^{4} + 749559 T^{5} + 7606351 T^{6} + 66930571 T^{7} + 568569433 T^{8} + 4374912530 T^{9} + 32506944020 T^{10} + 224232804625 T^{11} + 1496839126988 T^{12} + 9420655790867 T^{13} + 57528426652448 T^{14} + 334706601131190 T^{15} + 1894426983894998 T^{16} + 10292823012927236 T^{17} + 4957759867922089 p T^{18} + 278937081965414377 T^{19} + 1394357590223310396 T^{20} + 6757951985893357963 T^{21} + 32073522268231484092 T^{22} +$$$$14\!\cdots\!67$$$$T^{23} +$$$$67\!\cdots\!68$$$$T^{24} +$$$$29\!\cdots\!68$$$$T^{25} +$$$$12\!\cdots\!88$$$$T^{26} +$$$$54\!\cdots\!15$$$$T^{27} +$$$$22\!\cdots\!66$$$$T^{28} +$$$$93\!\cdots\!21$$$$T^{29} +$$$$37\!\cdots\!57$$$$T^{30} +$$$$14\!\cdots\!36$$$$T^{31} +$$$$58\!\cdots\!83$$$$T^{32} +$$$$22\!\cdots\!91$$$$T^{33} +$$$$76\!\cdots\!86$$$$p T^{34} +$$$$31\!\cdots\!26$$$$T^{35} +$$$$11\!\cdots\!72$$$$T^{36} +$$$$42\!\cdots\!41$$$$T^{37} +$$$$15\!\cdots\!21$$$$T^{38} +$$$$53\!\cdots\!82$$$$T^{39} +$$$$18\!\cdots\!85$$$$T^{40} +$$$$64\!\cdots\!86$$$$T^{41} +$$$$22\!\cdots\!31$$$$T^{42} +$$$$74\!\cdots\!74$$$$T^{43} +$$$$24\!\cdots\!80$$$$T^{44} +$$$$74\!\cdots\!74$$$$p T^{45} +$$$$22\!\cdots\!31$$$$p^{2} T^{46} +$$$$64\!\cdots\!86$$$$p^{3} T^{47} +$$$$18\!\cdots\!85$$$$p^{4} T^{48} +$$$$53\!\cdots\!82$$$$p^{5} T^{49} +$$$$15\!\cdots\!21$$$$p^{6} T^{50} +$$$$42\!\cdots\!41$$$$p^{7} T^{51} +$$$$11\!\cdots\!72$$$$p^{8} T^{52} +$$$$31\!\cdots\!26$$$$p^{9} T^{53} +$$$$76\!\cdots\!86$$$$p^{11} T^{54} +$$$$22\!\cdots\!91$$$$p^{11} T^{55} +$$$$58\!\cdots\!83$$$$p^{12} T^{56} +$$$$14\!\cdots\!36$$$$p^{13} T^{57} +$$$$37\!\cdots\!57$$$$p^{14} T^{58} +$$$$93\!\cdots\!21$$$$p^{15} T^{59} +$$$$22\!\cdots\!66$$$$p^{16} T^{60} +$$$$54\!\cdots\!15$$$$p^{17} T^{61} +$$$$12\!\cdots\!88$$$$p^{18} T^{62} +$$$$29\!\cdots\!68$$$$p^{19} T^{63} +$$$$67\!\cdots\!68$$$$p^{20} T^{64} +$$$$14\!\cdots\!67$$$$p^{21} T^{65} + 32073522268231484092 p^{22} T^{66} + 6757951985893357963 p^{23} T^{67} + 1394357590223310396 p^{24} T^{68} + 278937081965414377 p^{25} T^{69} + 4957759867922089 p^{27} T^{70} + 10292823012927236 p^{27} T^{71} + 1894426983894998 p^{28} T^{72} + 334706601131190 p^{29} T^{73} + 57528426652448 p^{30} T^{74} + 9420655790867 p^{31} T^{75} + 1496839126988 p^{32} T^{76} + 224232804625 p^{33} T^{77} + 32506944020 p^{34} T^{78} + 4374912530 p^{35} T^{79} + 568569433 p^{36} T^{80} + 66930571 p^{37} T^{81} + 7606351 p^{38} T^{82} + 749559 p^{39} T^{83} + 71734 p^{40} T^{84} + 494 p^{42} T^{85} + 411 p^{42} T^{86} + 19 p^{43} T^{87} + p^{44} T^{88}$$
13 $$1 + 10 T + 327 T^{2} + 2958 T^{3} + 53302 T^{4} + 441198 T^{5} + 5775959 T^{6} + 44173322 T^{7} + 468228261 T^{8} + 256582689 p T^{9} + 30294814573 T^{10} + 202407785530 T^{11} + 1629852313641 T^{12} + 10272073599449 T^{13} + 74999756608094 T^{14} + 448067790304288 T^{15} + 3013350706601434 T^{16} + 17136178487055052 T^{17} + 107377254649627389 T^{18} + 583309165672082099 T^{19} + 264251267272643454 p T^{20} + 17880819679305074702 T^{21} + 99640915898738130973 T^{22} +$$$$49\!\cdots\!41$$$$T^{23} +$$$$26\!\cdots\!43$$$$T^{24} +$$$$12\!\cdots\!06$$$$T^{25} +$$$$64\!\cdots\!71$$$$T^{26} +$$$$29\!\cdots\!87$$$$T^{27} +$$$$14\!\cdots\!47$$$$T^{28} +$$$$65\!\cdots\!96$$$$T^{29} +$$$$30\!\cdots\!19$$$$T^{30} +$$$$13\!\cdots\!38$$$$T^{31} +$$$$59\!\cdots\!02$$$$T^{32} +$$$$24\!\cdots\!81$$$$T^{33} +$$$$10\!\cdots\!18$$$$T^{34} +$$$$44\!\cdots\!88$$$$T^{35} +$$$$18\!\cdots\!01$$$$T^{36} +$$$$73\!\cdots\!56$$$$T^{37} +$$$$29\!\cdots\!95$$$$T^{38} +$$$$11\!\cdots\!50$$$$T^{39} +$$$$44\!\cdots\!06$$$$T^{40} +$$$$16\!\cdots\!83$$$$T^{41} +$$$$49\!\cdots\!67$$$$p T^{42} +$$$$23\!\cdots\!05$$$$T^{43} +$$$$85\!\cdots\!49$$$$T^{44} +$$$$23\!\cdots\!05$$$$p T^{45} +$$$$49\!\cdots\!67$$$$p^{3} T^{46} +$$$$16\!\cdots\!83$$$$p^{3} T^{47} +$$$$44\!\cdots\!06$$$$p^{4} T^{48} +$$$$11\!\cdots\!50$$$$p^{5} T^{49} +$$$$29\!\cdots\!95$$$$p^{6} T^{50} +$$$$73\!\cdots\!56$$$$p^{7} T^{51} +$$$$18\!\cdots\!01$$$$p^{8} T^{52} +$$$$44\!\cdots\!88$$$$p^{9} T^{53} +$$$$10\!\cdots\!18$$$$p^{10} T^{54} +$$$$24\!\cdots\!81$$$$p^{11} T^{55} +$$$$59\!\cdots\!02$$$$p^{12} T^{56} +$$$$13\!\cdots\!38$$$$p^{13} T^{57} +$$$$30\!\cdots\!19$$$$p^{14} T^{58} +$$$$65\!\cdots\!96$$$$p^{15} T^{59} +$$$$14\!\cdots\!47$$$$p^{16} T^{60} +$$$$29\!\cdots\!87$$$$p^{17} T^{61} +$$$$64\!\cdots\!71$$$$p^{18} T^{62} +$$$$12\!\cdots\!06$$$$p^{19} T^{63} +$$$$26\!\cdots\!43$$$$p^{20} T^{64} +$$$$49\!\cdots\!41$$$$p^{21} T^{65} + 99640915898738130973 p^{22} T^{66} + 17880819679305074702 p^{23} T^{67} + 264251267272643454 p^{25} T^{68} + 583309165672082099 p^{25} T^{69} + 107377254649627389 p^{26} T^{70} + 17136178487055052 p^{27} T^{71} + 3013350706601434 p^{28} T^{72} + 448067790304288 p^{29} T^{73} + 74999756608094 p^{30} T^{74} + 10272073599449 p^{31} T^{75} + 1629852313641 p^{32} T^{76} + 202407785530 p^{33} T^{77} + 30294814573 p^{34} T^{78} + 256582689 p^{36} T^{79} + 468228261 p^{36} T^{80} + 44173322 p^{37} T^{81} + 5775959 p^{38} T^{82} + 441198 p^{39} T^{83} + 53302 p^{40} T^{84} + 2958 p^{41} T^{85} + 327 p^{42} T^{86} + 10 p^{43} T^{87} + p^{44} T^{88}$$
17 $$1 + 16 T + 500 T^{2} + 6428 T^{3} + 6845 p T^{4} + 1270341 T^{5} + 17222018 T^{6} + 165203854 T^{7} + 108748486 p T^{8} + 15946118972 T^{9} + 154869164114 T^{10} + 1221105998952 T^{11} + 10607911873978 T^{12} + 4553267000701 p T^{13} + 613643161014672 T^{14} + 4183538214487573 T^{15} + 30699453813107487 T^{16} + 197026062960696174 T^{17} + 1352564373468714023 T^{18} + 8221999014222713292 T^{19} + 53238143798083018787 T^{20} +$$$$30\!\cdots\!07$$$$T^{21} +$$$$18\!\cdots\!25$$$$T^{22} +$$$$10\!\cdots\!93$$$$T^{23} +$$$$36\!\cdots\!91$$$$p T^{24} +$$$$32\!\cdots\!66$$$$T^{25} +$$$$18\!\cdots\!49$$$$T^{26} +$$$$93\!\cdots\!32$$$$T^{27} +$$$$50\!\cdots\!82$$$$T^{28} +$$$$25\!\cdots\!20$$$$T^{29} +$$$$13\!\cdots\!36$$$$T^{30} +$$$$62\!\cdots\!03$$$$T^{31} +$$$$31\!\cdots\!42$$$$T^{32} +$$$$14\!\cdots\!85$$$$T^{33} +$$$$71\!\cdots\!34$$$$T^{34} +$$$$32\!\cdots\!74$$$$T^{35} +$$$$90\!\cdots\!06$$$$p T^{36} +$$$$67\!\cdots\!04$$$$T^{37} +$$$$31\!\cdots\!04$$$$T^{38} +$$$$13\!\cdots\!36$$$$T^{39} +$$$$59\!\cdots\!45$$$$T^{40} +$$$$14\!\cdots\!24$$$$p T^{41} +$$$$64\!\cdots\!53$$$$p T^{42} +$$$$44\!\cdots\!53$$$$T^{43} +$$$$19\!\cdots\!44$$$$T^{44} +$$$$44\!\cdots\!53$$$$p T^{45} +$$$$64\!\cdots\!53$$$$p^{3} T^{46} +$$$$14\!\cdots\!24$$$$p^{4} T^{47} +$$$$59\!\cdots\!45$$$$p^{4} T^{48} +$$$$13\!\cdots\!36$$$$p^{5} T^{49} +$$$$31\!\cdots\!04$$$$p^{6} T^{50} +$$$$67\!\cdots\!04$$$$p^{7} T^{51} +$$$$90\!\cdots\!06$$$$p^{9} T^{52} +$$$$32\!\cdots\!74$$$$p^{9} T^{53} +$$$$71\!\cdots\!34$$$$p^{10} T^{54} +$$$$14\!\cdots\!85$$$$p^{11} T^{55} +$$$$31\!\cdots\!42$$$$p^{12} T^{56} +$$$$62\!\cdots\!03$$$$p^{13} T^{57} +$$$$13\!\cdots\!36$$$$p^{14} T^{58} +$$$$25\!\cdots\!20$$$$p^{15} T^{59} +$$$$50\!\cdots\!82$$$$p^{16} T^{60} +$$$$93\!\cdots\!32$$$$p^{17} T^{61} +$$$$18\!\cdots\!49$$$$p^{18} T^{62} +$$$$32\!\cdots\!66$$$$p^{19} T^{63} +$$$$36\!\cdots\!91$$$$p^{21} T^{64} +$$$$10\!\cdots\!93$$$$p^{21} T^{65} +$$$$18\!\cdots\!25$$$$p^{22} T^{66} +$$$$30\!\cdots\!07$$$$p^{23} T^{67} + 53238143798083018787 p^{24} T^{68} + 8221999014222713292 p^{25} T^{69} + 1352564373468714023 p^{26} T^{70} + 197026062960696174 p^{27} T^{71} + 30699453813107487 p^{28} T^{72} + 4183538214487573 p^{29} T^{73} + 613643161014672 p^{30} T^{74} + 4553267000701 p^{32} T^{75} + 10607911873978 p^{32} T^{76} + 1221105998952 p^{33} T^{77} + 154869164114 p^{34} T^{78} + 15946118972 p^{35} T^{79} + 108748486 p^{37} T^{80} + 165203854 p^{37} T^{81} + 17222018 p^{38} T^{82} + 1270341 p^{39} T^{83} + 6845 p^{41} T^{84} + 6428 p^{41} T^{85} + 500 p^{42} T^{86} + 16 p^{43} T^{87} + p^{44} T^{88}$$
19 $$1 + 25 T + 747 T^{2} + 13265 T^{3} + 241810 T^{4} + 3433362 T^{5} + 48204457 T^{6} + 580895669 T^{7} + 6843299099 T^{8} + 72508922446 T^{9} + 748621008198 T^{10} + 7137171995718 T^{11} + 66284437022011 T^{12} + 577835378910278 T^{13} + 4911149408770784 T^{14} + 39611293431874685 T^{15} + 311875825259324725 T^{16} + 2348259246568232260 T^{17} + 17283183039524474131 T^{18} +$$$$12\!\cdots\!83$$$$T^{19} +$$$$84\!\cdots\!78$$$$T^{20} +$$$$56\!\cdots\!99$$$$T^{21} +$$$$37\!\cdots\!26$$$$T^{22} +$$$$23\!\cdots\!46$$$$T^{23} +$$$$14\!\cdots\!45$$$$T^{24} +$$$$89\!\cdots\!60$$$$T^{25} +$$$$53\!\cdots\!57$$$$T^{26} +$$$$30\!\cdots\!54$$$$T^{27} +$$$$17\!\cdots\!73$$$$T^{28} +$$$$97\!\cdots\!38$$$$T^{29} +$$$$53\!\cdots\!64$$$$T^{30} +$$$$28\!\cdots\!00$$$$T^{31} +$$$$15\!\cdots\!77$$$$T^{32} +$$$$78\!\cdots\!00$$$$T^{33} +$$$$39\!\cdots\!88$$$$T^{34} +$$$$19\!\cdots\!23$$$$T^{35} +$$$$98\!\cdots\!80$$$$T^{36} +$$$$47\!\cdots\!80$$$$T^{37} +$$$$22\!\cdots\!67$$$$T^{38} +$$$$56\!\cdots\!57$$$$p T^{39} +$$$$49\!\cdots\!21$$$$T^{40} +$$$$22\!\cdots\!51$$$$T^{41} +$$$$10\!\cdots\!02$$$$T^{42} +$$$$45\!\cdots\!73$$$$T^{43} +$$$$20\!\cdots\!02$$$$T^{44} +$$$$45\!\cdots\!73$$$$p T^{45} +$$$$10\!\cdots\!02$$$$p^{2} T^{46} +$$$$22\!\cdots\!51$$$$p^{3} T^{47} +$$$$49\!\cdots\!21$$$$p^{4} T^{48} +$$$$56\!\cdots\!57$$$$p^{6} T^{49} +$$$$22\!\cdots\!67$$$$p^{6} T^{50} +$$$$47\!\cdots\!80$$$$p^{7} T^{51} +$$$$98\!\cdots\!80$$$$p^{8} T^{52} +$$$$19\!\cdots\!23$$$$p^{9} T^{53} +$$$$39\!\cdots\!88$$$$p^{10} T^{54} +$$$$78\!\cdots\!00$$$$p^{11} T^{55} +$$$$15\!\cdots\!77$$$$p^{12} T^{56} +$$$$28\!\cdots\!00$$$$p^{13} T^{57} +$$$$53\!\cdots\!64$$$$p^{14} T^{58} +$$$$97\!\cdots\!38$$$$p^{15} T^{59} +$$$$17\!\cdots\!73$$$$p^{16} T^{60} +$$$$30\!\cdots\!54$$$$p^{17} T^{61} +$$$$53\!\cdots\!57$$$$p^{18} T^{62} +$$$$89\!\cdots\!60$$$$p^{19} T^{63} +$$$$14\!\cdots\!45$$$$p^{20} T^{64} +$$$$23\!\cdots\!46$$$$p^{21} T^{65} +$$$$37\!\cdots\!26$$$$p^{22} T^{66} +$$$$56\!\cdots\!99$$$$p^{23} T^{67} +$$$$84\!\cdots\!78$$$$p^{24} T^{68} +$$$$12\!\cdots\!83$$$$p^{25} T^{69} + 17283183039524474131 p^{26} T^{70} + 2348259246568232260 p^{27} T^{71} + 311875825259324725 p^{28} T^{72} + 39611293431874685 p^{29} T^{73} + 4911149408770784 p^{30} T^{74} + 577835378910278 p^{31} T^{75} + 66284437022011 p^{32} T^{76} + 7137171995718 p^{33} T^{77} + 748621008198 p^{34} T^{78} + 72508922446 p^{35} T^{79} + 6843299099 p^{36} T^{80} + 580895669 p^{37} T^{81} + 48204457 p^{38} T^{82} + 3433362 p^{39} T^{83} + 241810 p^{40} T^{84} + 13265 p^{41} T^{85} + 747 p^{42} T^{86} + 25 p^{43} T^{87} + p^{44} T^{88}$$
23 $$1 + 29 T + 891 T^{2} + 758 p T^{3} + 330440 T^{4} + 5060360 T^{5} + 74137876 T^{6} + 952735960 T^{7} + 11723966224 T^{8} + 131583176757 T^{9} + 1419394748301 T^{10} + 14275507710249 T^{11} + 6023847712452 p T^{12} + 1271253950576027 T^{13} + 11297950328590908 T^{14} + 95833201330456179 T^{15} + 789956535771934355 T^{16} + 6258095056709779354 T^{17} + 2100802881717419016 p T^{18} +$$$$36\!\cdots\!16$$$$T^{19} +$$$$26\!\cdots\!11$$$$T^{20} +$$$$18\!\cdots\!29$$$$T^{21} +$$$$12\!\cdots\!00$$$$T^{22} +$$$$86\!\cdots\!37$$$$T^{23} +$$$$57\!\cdots\!51$$$$T^{24} +$$$$37\!\cdots\!02$$$$T^{25} +$$$$23\!\cdots\!70$$$$T^{26} +$$$$14\!\cdots\!78$$$$T^{27} +$$$$89\!\cdots\!16$$$$T^{28} +$$$$53\!\cdots\!04$$$$T^{29} +$$$$31\!\cdots\!79$$$$T^{30} +$$$$18\!\cdots\!45$$$$T^{31} +$$$$10\!\cdots\!83$$$$T^{32} +$$$$58\!\cdots\!27$$$$T^{33} +$$$$32\!\cdots\!23$$$$T^{34} +$$$$17\!\cdots\!85$$$$T^{35} +$$$$95\!\cdots\!49$$$$T^{36} +$$$$50\!\cdots\!98$$$$T^{37} +$$$$26\!\cdots\!19$$$$T^{38} +$$$$13\!\cdots\!13$$$$T^{39} +$$$$69\!\cdots\!12$$$$T^{40} +$$$$34\!\cdots\!73$$$$T^{41} +$$$$17\!\cdots\!49$$$$T^{42} +$$$$84\!\cdots\!44$$$$T^{43} +$$$$40\!\cdots\!64$$$$T^{44} +$$$$84\!\cdots\!44$$$$p T^{45} +$$$$17\!\cdots\!49$$$$p^{2} T^{46} +$$$$34\!\cdots\!73$$$$p^{3} T^{47} +$$$$69\!\cdots\!12$$$$p^{4} T^{48} +$$$$13\!\cdots\!13$$$$p^{5} T^{49} +$$$$26\!\cdots\!19$$$$p^{6} T^{50} +$$$$50\!\cdots\!98$$$$p^{7} T^{51} +$$$$95\!\cdots\!49$$$$p^{8} T^{52} +$$$$17\!\cdots\!85$$$$p^{9} T^{53} +$$$$32\!\cdots\!23$$$$p^{10} T^{54} +$$$$58\!\cdots\!27$$$$p^{11} T^{55} +$$$$10\!\cdots\!83$$$$p^{12} T^{56} +$$$$18\!\cdots\!45$$$$p^{13} T^{57} +$$$$31\!\cdots\!79$$$$p^{14} T^{58} +$$$$53\!\cdots\!04$$$$p^{15} T^{59} +$$$$89\!\cdots\!16$$$$p^{16} T^{60} +$$$$14\!\cdots\!78$$$$p^{17} T^{61} +$$$$23\!\cdots\!70$$$$p^{18} T^{62} +$$$$37\!\cdots\!02$$$$p^{19} T^{63} +$$$$57\!\cdots\!51$$$$p^{20} T^{64} +$$$$86\!\cdots\!37$$$$p^{21} T^{65} +$$$$12\!\cdots\!00$$$$p^{22} T^{66} +$$$$18\!\cdots\!29$$$$p^{23} T^{67} +$$$$26\!\cdots\!11$$$$p^{24} T^{68} +$$$$36\!\cdots\!16$$$$p^{25} T^{69} + 2100802881717419016 p^{27} T^{70} + 6258095056709779354 p^{27} T^{71} + 789956535771934355 p^{28} T^{72} + 95833201330456179 p^{29} T^{73} + 11297950328590908 p^{30} T^{74} + 1271253950576027 p^{31} T^{75} + 6023847712452 p^{33} T^{76} + 14275507710249 p^{33} T^{77} + 1419394748301 p^{34} T^{78} + 131583176757 p^{35} T^{79} + 11723966224 p^{36} T^{80} + 952735960 p^{37} T^{81} + 74137876 p^{38} T^{82} + 5060360 p^{39} T^{83} + 330440 p^{40} T^{84} + 758 p^{42} T^{85} + 891 p^{42} T^{86} + 29 p^{43} T^{87} + p^{44} T^{88}$$
29 $$1 - 35 T + 1233 T^{2} - 28396 T^{3} + 618260 T^{4} - 11030120 T^{5} + 185647347 T^{6} - 2765511577 T^{7} + 39101602017 T^{8} - 507561378041 T^{9} + 217071500101 p T^{10} - 73178431041204 T^{11} + 817545945771973 T^{12} - 8673476119381736 T^{13} + 88863004297160569 T^{14} - 872297290828174867 T^{15} + 8301815410145625889 T^{16} - 76183928007844506527 T^{17} +$$$$68\!\cdots\!20$$$$T^{18} -$$$$58\!\cdots\!08$$$$T^{19} +$$$$49\!\cdots\!03$$$$T^{20} -$$$$40\!\cdots\!66$$$$T^{21} +$$$$32\!\cdots\!58$$$$T^{22} -$$$$25\!\cdots\!90$$$$T^{23} +$$$$19\!\cdots\!56$$$$T^{24} -$$$$50\!\cdots\!59$$$$p T^{25} +$$$$10\!\cdots\!77$$$$T^{26} -$$$$76\!\cdots\!77$$$$T^{27} +$$$$54\!\cdots\!82$$$$T^{28} -$$$$37\!\cdots\!42$$$$T^{29} +$$$$25\!\cdots\!31$$$$T^{30} -$$$$16\!\cdots\!06$$$$T^{31} +$$$$11\!\cdots\!07$$$$T^{32} -$$$$71\!\cdots\!83$$$$T^{33} +$$$$45\!\cdots\!21$$$$T^{34} -$$$$28\!\cdots\!36$$$$T^{35} +$$$$17\!\cdots\!69$$$$T^{36} -$$$$10\!\cdots\!48$$$$T^{37} +$$$$61\!\cdots\!05$$$$T^{38} -$$$$36\!\cdots\!22$$$$T^{39} +$$$$20\!\cdots\!66$$$$T^{40} -$$$$11\!\cdots\!45$$$$T^{41} +$$$$66\!\cdots\!42$$$$T^{42} -$$$$36\!\cdots\!91$$$$T^{43} +$$$$19\!\cdots\!98$$$$T^{44} -$$$$36\!\cdots\!91$$$$p T^{45} +$$$$66\!\cdots\!42$$$$p^{2} T^{46} -$$$$11\!\cdots\!45$$$$p^{3} T^{47} +$$$$20\!\cdots\!66$$$$p^{4} T^{48} -$$$$36\!\cdots\!22$$$$p^{5} T^{49} +$$$$61\!\cdots\!05$$$$p^{6} T^{50} -$$$$10\!\cdots\!48$$$$p^{7} T^{51} +$$$$17\!\cdots\!69$$$$p^{8} T^{52} -$$$$28\!\cdots\!36$$$$p^{9} T^{53} +$$$$45\!\cdots\!21$$$$p^{10} T^{54} -$$$$71\!\cdots\!83$$$$p^{11} T^{55} +$$$$11\!\cdots\!07$$$$p^{12} T^{56} -$$$$16\!\cdots\!06$$$$p^{13} T^{57} +$$$$25\!\cdots\!31$$$$p^{14} T^{58} -$$$$37\!\cdots\!42$$$$p^{15} T^{59} +$$$$54\!\cdots\!82$$$$p^{16} T^{60} -$$$$76\!\cdots\!77$$$$p^{17} T^{61} +$$$$10\!\cdots\!77$$$$p^{18} T^{62} -$$$$50\!\cdots\!59$$$$p^{20} T^{63} +$$$$19\!\cdots\!56$$$$p^{20} T^{64} -$$$$25\!\cdots\!90$$$$p^{21} T^{65} +$$$$32\!\cdots\!58$$$$p^{22} T^{66} -$$$$40\!\cdots\!66$$$$p^{23} T^{67} +$$$$49\!\cdots\!03$$$$p^{24} T^{68} -$$$$58\!\cdots\!08$$$$p^{25} T^{69} +$$$$68\!\cdots\!20$$$$p^{26} T^{70} - 76183928007844506527 p^{27} T^{71} + 8301815410145625889 p^{28} T^{72} - 872297290828174867 p^{29} T^{73} + 88863004297160569 p^{30} T^{74} - 8673476119381736 p^{31} T^{75} + 817545945771973 p^{32} T^{76} - 73178431041204 p^{33} T^{77} + 217071500101 p^{35} T^{78} - 507561378041 p^{35} T^{79} + 39101602017 p^{36} T^{80} - 2765511577 p^{37} T^{81} + 185647347 p^{38} T^{82} - 11030120 p^{39} T^{83} + 618260 p^{40} T^{84} - 28396 p^{41} T^{85} + 1233 p^{42} T^{86} - 35 p^{43} T^{87} + p^{44} T^{88}$$
31 $$1 + 49 T + 1925 T^{2} + 55297 T^{3} + 1378487 T^{4} + 29621401 T^{5} + 575763120 T^{6} + 10166721076 T^{7} + 166321987289 T^{8} + 2533758269015 T^{9} + 36305427816147 T^{10} + 491387665829229 T^{11} + 6319722976122190 T^{12} + 77490869408168447 T^{13} + 909448754835801697 T^{14} + 330412233750271793 p T^{15} +$$$$11\!\cdots\!35$$$$T^{16} + 37426170282699275378 p T^{17} +$$$$11\!\cdots\!89$$$$T^{18} +$$$$11\!\cdots\!43$$$$T^{19} +$$$$10\!\cdots\!79$$$$T^{20} +$$$$99\!\cdots\!92$$$$T^{21} +$$$$89\!\cdots\!99$$$$T^{22} +$$$$77\!\cdots\!15$$$$T^{23} +$$$$65\!\cdots\!48$$$$T^{24} +$$$$54\!\cdots\!17$$$$T^{25} +$$$$43\!\cdots\!05$$$$T^{26} +$$$$34\!\cdots\!66$$$$T^{27} +$$$$26\!\cdots\!14$$$$T^{28} +$$$$19\!\cdots\!52$$$$T^{29} +$$$$14\!\cdots\!45$$$$T^{30} +$$$$10\!\cdots\!74$$$$T^{31} +$$$$74\!\cdots\!36$$$$T^{32} +$$$$51\!\cdots\!76$$$$T^{33} +$$$$34\!\cdots\!20$$$$T^{34} +$$$$23\!\cdots\!74$$$$T^{35} +$$$$15\!\cdots\!40$$$$T^{36} +$$$$96\!\cdots\!23$$$$T^{37} +$$$$60\!\cdots\!31$$$$T^{38} +$$$$37\!\cdots\!18$$$$T^{39} +$$$$22\!\cdots\!75$$$$T^{40} +$$$$13\!\cdots\!95$$$$T^{41} +$$$$77\!\cdots\!02$$$$T^{42} +$$$$44\!\cdots\!50$$$$T^{43} +$$$$24\!\cdots\!12$$$$T^{44} +$$$$44\!\cdots\!50$$$$p T^{45} +$$$$77\!\cdots\!02$$$$p^{2} T^{46} +$$$$13\!\cdots\!95$$$$p^{3} T^{47} +$$$$22\!\cdots\!75$$$$p^{4} T^{48} +$$$$37\!\cdots\!18$$$$p^{5} T^{49} +$$$$60\!\cdots\!31$$$$p^{6} T^{50} +$$$$96\!\cdots\!23$$$$p^{7} T^{51} +$$$$15\!\cdots\!40$$$$p^{8} T^{52} +$$$$23\!\cdots\!74$$$$p^{9} T^{53} +$$$$34\!\cdots\!20$$$$p^{10} T^{54} +$$$$51\!\cdots\!76$$$$p^{11} T^{55} +$$$$74\!\cdots\!36$$$$p^{12} T^{56} +$$$$10\!\cdots\!74$$$$p^{13} T^{57} +$$$$14\!\cdots\!45$$$$p^{14} T^{58} +$$$$19\!\cdots\!52$$$$p^{15} T^{59} +$$$$26\!\cdots\!14$$$$p^{16} T^{60} +$$$$34\!\cdots\!66$$$$p^{17} T^{61} +$$$$43\!\cdots\!05$$$$p^{18} T^{62} +$$$$54\!\cdots\!17$$$$p^{19} T^{63} +$$$$65\!\cdots\!48$$$$p^{20} T^{64} +$$$$77\!\cdots\!15$$$$p^{21} T^{65} +$$$$89\!\cdots\!99$$$$p^{22} T^{66} +$$$$99\!\cdots\!92$$$$p^{23} T^{67} +$$$$10\!\cdots\!79$$$$p^{24} T^{68} +$$$$11\!\cdots\!43$$$$p^{25} T^{69} +$$$$11\!\cdots\!89$$$$p^{26} T^{70} + 37426170282699275378 p^{28} T^{71} +$$$$11\!\cdots\!35$$$$p^{28} T^{72} + 330412233750271793 p^{30} T^{73} + 909448754835801697 p^{30} T^{74} + 77490869408168447 p^{31} T^{75} + 6319722976122190 p^{32} T^{76} + 491387665829229 p^{33} T^{77} + 36305427816147 p^{34} T^{78} + 2533758269015 p^{35} T^{79} + 166321987289 p^{36} T^{80} + 10166721076 p^{37} T^{81} + 575763120 p^{38} T^{82} + 29621401 p^{39} T^{83} + 1378487 p^{40} T^{84} + 55297 p^{41} T^{85} + 1925 p^{42} T^{86} + 49 p^{43} T^{87} + p^{44} T^{88}$$
37 $$1 + 30 T + 1243 T^{2} + 27648 T^{3} + 18642 p T^{4} + 12503067 T^{5} + 238534889 T^{6} + 3704775316 T^{7} + 59019504202 T^{8} + 21883992359 p T^{9} + 11260882597756 T^{10} + 139250886237094 T^{11} + 1735934646315917 T^{12} + 19625006242979369 T^{13} + 223140841084842060 T^{14} + 2330297119632329654 T^{15} + 24463634223868384932 T^{16} + 6428369817287760566 p T^{17} +$$$$23\!\cdots\!33$$$$T^{18} +$$$$21\!\cdots\!16$$$$T^{19} +$$$$19\!\cdots\!57$$$$T^{20} +$$$$16\!\cdots\!43$$$$T^{21} +$$$$14\!\cdots\!12$$$$T^{22} +$$$$11\!\cdots\!59$$$$T^{23} +$$$$95\!\cdots\!65$$$$T^{24} +$$$$73\!\cdots\!21$$$$T^{25} +$$$$57\!\cdots\!74$$$$T^{26} +$$$$41\!\cdots\!57$$$$T^{27} +$$$$31\!\cdots\!15$$$$T^{28} +$$$$21\!\cdots\!18$$$$T^{29} +$$$$15\!\cdots\!65$$$$T^{30} +$$$$10\!\cdots\!96$$$$T^{31} +$$$$70\!\cdots\!29$$$$T^{32} +$$$$44\!\cdots\!44$$$$T^{33} +$$$$29\!\cdots\!03$$$$T^{34} +$$$$17\!\cdots\!66$$$$T^{35} +$$$$11\!\cdots\!92$$$$T^{36} +$$$$67\!\cdots\!28$$$$T^{37} +$$$$43\!\cdots\!65$$$$T^{38} +$$$$24\!\cdots\!03$$$$T^{39} +$$$$15\!\cdots\!46$$$$T^{40} +$$$$86\!\cdots\!42$$$$T^{41} +$$$$56\!\cdots\!36$$$$T^{42} +$$$$30\!\cdots\!80$$$$T^{43} +$$$$20\!\cdots\!60$$$$T^{44} +$$$$30\!\cdots\!80$$$$p T^{45} +$$$$56\!\cdots\!36$$$$p^{2} T^{46} +$$$$86\!\cdots\!42$$$$p^{3} T^{47} +$$$$15\!\cdots\!46$$$$p^{4} T^{48} +$$$$24\!\cdots\!03$$$$p^{5} T^{49} +$$$$43\!\cdots\!65$$$$p^{6} T^{50} +$$$$67\!\cdots\!28$$$$p^{7} T^{51} +$$$$11\!\cdots\!92$$$$p^{8} T^{52} +$$$$17\!\cdots\!66$$$$p^{9} T^{53} +$$$$29\!\cdots\!03$$$$p^{10} T^{54} +$$$$44\!\cdots\!44$$$$p^{11} T^{55} +$$$$70\!\cdots\!29$$$$p^{12} T^{56} +$$$$10\!\cdots\!96$$$$p^{13} T^{57} +$$$$15\!\cdots\!65$$$$p^{14} T^{58} +$$$$21\!\cdots\!18$$$$p^{15} T^{59} +$$$$31\!\cdots\!15$$$$p^{16} T^{60} +$$$$41\!\cdots\!57$$$$p^{17} T^{61} +$$$$57\!\cdots\!74$$$$p^{18} T^{62} +$$$$73\!\cdots\!21$$$$p^{19} T^{63} +$$$$95\!\cdots\!65$$$$p^{20} T^{64} +$$$$11\!\cdots\!59$$$$p^{21} T^{65} +$$$$14\!\cdots\!12$$$$p^{22} T^{66} +$$$$16\!\cdots\!43$$$$p^{23} T^{67} +$$$$19\!\cdots\!57$$$$p^{24} T^{68} +$$$$21\!\cdots\!16$$$$p^{25} T^{69} +$$$$23\!\cdots\!33$$$$p^{26} T^{70} + 6428369817287760566 p^{28} T^{71} + 24463634223868384932 p^{28} T^{72} + 2330297119632329654 p^{29} T^{73} + 223140841084842060 p^{30} T^{74} + 19625006242979369 p^{31} T^{75} + 1735934646315917 p^{32} T^{76} + 139250886237094 p^{33} T^{77} + 11260882597756 p^{34} T^{78} + 21883992359 p^{36} T^{79} + 59019504202 p^{36} T^{80} + 3704775316 p^{37} T^{81} + 238534889 p^{38} T^{82} + 12503067 p^{39} T^{83} + 18642 p^{41} T^{84} + 27648 p^{41} T^{85} + 1243 p^{42} T^{86} + 30 p^{43} T^{87} + p^{44} T^{88}$$
41 $$1 + 14 T + 1130 T^{2} + 13684 T^{3} + 614861 T^{4} + 159273 p T^{5} + 216270954 T^{6} + 2031486298 T^{7} + 55622190930 T^{8} + 464107426894 T^{9} + 11208944783734 T^{10} + 83164538627006 T^{11} + 1851291770615317 T^{12} + 12190913884912900 T^{13} + 258734312927412465 T^{14} + 1505276656903067343 T^{15} + 31343402811538996016 T^{16} +$$$$15\!\cdots\!87$$$$T^{17} +$$$$33\!\cdots\!14$$$$T^{18} +$$$$14\!\cdots\!81$$$$T^{19} +$$$$32\!\cdots\!83$$$$T^{20} +$$$$12\!\cdots\!45$$$$T^{21} +$$$$27\!\cdots\!15$$$$T^{22} +$$$$89\!\cdots\!48$$$$T^{23} +$$$$22\!\cdots\!36$$$$T^{24} +$$$$59\!\cdots\!51$$$$T^{25} +$$$$16\!\cdots\!67$$$$T^{26} +$$$$35\!\cdots\!42$$$$T^{27} +$$$$11\!\cdots\!39$$$$T^{28} +$$$$19\!\cdots\!71$$$$T^{29} +$$$$71\!\cdots\!77$$$$T^{30} +$$$$94\!\cdots\!49$$$$T^{31} +$$$$42\!\cdots\!61$$$$T^{32} +$$$$42\!\cdots\!96$$$$T^{33} +$$$$24\!\cdots\!39$$$$T^{34} +$$$$16\!\cdots\!92$$$$T^{35} +$$$$12\!\cdots\!96$$$$T^{36} +$$$$60\!\cdots\!41$$$$T^{37} +$$$$63\!\cdots\!52$$$$T^{38} +$$$$19\!\cdots\!73$$$$T^{39} +$$$$30\!\cdots\!68$$$$T^{40} +$$$$60\!\cdots\!55$$$$T^{41} +$$$$13\!\cdots\!61$$$$T^{42} +$$$$19\!\cdots\!53$$$$T^{43} +$$$$56\!\cdots\!40$$$$T^{44} +$$$$19\!\cdots\!53$$$$p T^{45} +$$$$13\!\cdots\!61$$$$p^{2} T^{46} +$$$$60\!\cdots\!55$$$$p^{3} T^{47} +$$$$30\!\cdots\!68$$$$p^{4} T^{48} +$$$$19\!\cdots\!73$$$$p^{5} T^{49} +$$$$63\!\cdots\!52$$$$p^{6} T^{50} +$$$$60\!\cdots\!41$$$$p^{7} T^{51} +$$$$12\!\cdots\!96$$$$p^{8} T^{52} +$$$$16\!\cdots\!92$$$$p^{9} T^{53} +$$$$24\!\cdots\!39$$$$p^{10} T^{54} +$$$$42\!\cdots\!96$$$$p^{11} T^{55} +$$$$42\!\cdots\!61$$$$p^{12} T^{56} +$$$$94\!\cdots\!49$$$$p^{13} T^{57} +$$$$71\!\cdots\!77$$$$p^{14} T^{58} +$$$$19\!\cdots\!71$$$$p^{15} T^{59} +$$$$11\!\cdots\!39$$$$p^{16} T^{60} +$$$$35\!\cdots\!42$$$$p^{17} T^{61} +$$$$16\!\cdots\!67$$$$p^{18} T^{62} +$$$$59\!\cdots\!51$$$$p^{19} T^{63} +$$$$22\!\cdots\!36$$$$p^{20} T^{64} +$$$$89\!\cdots\!48$$$$p^{21} T^{65} +$$$$27\!\cdots\!15$$$$p^{22} T^{66} +$$$$12\!\cdots\!45$$$$p^{23} T^{67} +$$$$32\!\cdots\!83$$$$p^{24} T^{68} +$$$$14\!\cdots\!81$$$$p^{25} T^{69} +$$$$33\!\cdots\!14$$$$p^{26} T^{70} +$$$$15\!\cdots\!87$$$$p^{27} T^{71} + 31343402811538996016 p^{28} T^{72} + 1505276656903067343 p^{29} T^{73} + 258734312927412465 p^{30} T^{74} + 12190913884912900 p^{31} T^{75} + 1851291770615317 p^{32} T^{76} + 83164538627006 p^{33} T^{77} + 11208944783734 p^{34} T^{78} + 464107426894 p^{35} T^{79} + 55622190930 p^{36} T^{80} + 2031486298 p^{37} T^{81} + 216270954 p^{38} T^{82} + 159273 p^{40} T^{83} + 614861 p^{40} T^{84} + 13684 p^{41} T^{85} + 1130 p^{42} T^{86} + 14 p^{43} T^{87} + p^{44} T^{88}$$
43 $$1 + 35 T + 1626 T^{2} + 41836 T^{3} + 1173233 T^{4} + 24529122 T^{5} + 527429182 T^{6} + 9449040322 T^{7} + 170189860166 T^{8} + 2698739936154 T^{9} + 42591813101713 T^{10} + 610852036948918 T^{11} + 201812637620028 p T^{12} + 114297937396905555 T^{13} + 34600523118082039 p T^{14} + 18200099284451033495 T^{15} +$$$$21\!\cdots\!83$$$$T^{16} +$$$$25\!\cdots\!27$$$$T^{17} +$$$$28\!\cdots\!79$$$$T^{18} +$$$$30\!\cdots\!59$$$$T^{19} +$$$$32\!\cdots\!03$$$$T^{20} +$$$$33\!\cdots\!96$$$$T^{21} +$$$$34\!\cdots\!34$$$$T^{22} +$$$$33\!\cdots\!72$$$$T^{23} +$$$$31\!\cdots\!65$$$$T^{24} +$$$$29\!\cdots\!69$$$$T^{25} +$$$$27\!\cdots\!62$$$$T^{26} +$$$$24\!\cdots\!76$$$$T^{27} +$$$$21\!\cdots\!10$$$$T^{28} +$$$$18\!\cdots\!68$$$$T^{29} +$$$$15\!\cdots\!03$$$$T^{30} +$$$$12\!\cdots\!44$$$$T^{31} +$$$$10\!\cdots\!09$$$$T^{32} +$$$$83\!\cdots\!86$$$$T^{33} +$$$$66\!\cdots\!47$$$$T^{34} +$$$$50\!\cdots\!54$$$$T^{35} +$$$$38\!\cdots\!54$$$$T^{36} +$$$$28\!\cdots\!03$$$$T^{37} +$$$$20\!\cdots\!79$$$$T^{38} +$$$$14\!\cdots\!67$$$$T^{39} +$$$$10\!\cdots\!18$$$$T^{40} +$$$$73\!\cdots\!95$$$$T^{41} +$$$$50\!\cdots\!58$$$$T^{42} +$$$$33\!\cdots\!55$$$$T^{43} +$$$$22\!\cdots\!00$$$$T^{44} +$$$$33\!\cdots\!55$$$$p T^{45} +$$$$50\!\cdots\!58$$$$p^{2} T^{46} +$$$$73\!\cdots\!95$$$$p^{3} T^{47} +$$$$10\!\cdots\!18$$$$p^{4} T^{48} +$$$$14\!\cdots\!67$$$$p^{5} T^{49} +$$$$20\!\cdots\!79$$$$p^{6} T^{50} +$$$$28\!\cdots\!03$$$$p^{7} T^{51} +$$$$38\!\cdots\!54$$$$p^{8} T^{52} +$$$$50\!\cdots\!54$$$$p^{9} T^{53} +$$$$66\!\cdots\!47$$$$p^{10} T^{54} +$$$$83\!\cdots\!86$$$$p^{11} T^{55} +$$$$10\!\cdots\!09$$$$p^{12} T^{56} +$$$$12\!\cdots\!44$$$$p^{13} T^{57} +$$$$15\!\cdots\!03$$$$p^{14} T^{58} +$$$$18\!\cdots\!68$$$$p^{15} T^{59} +$$$$21\!\cdots\!10$$$$p^{16} T^{60} +$$$$24\!\cdots\!76$$$$p^{17} T^{61} +$$$$27\!\cdots\!62$$$$p^{18} T^{62} +$$$$29\!\cdots\!69$$$$p^{19} T^{63} +$$$$31\!\cdots\!65$$$$p^{20} T^{64} +$$$$33\!\cdots\!72$$$$p^{21} T^{65} +$$$$34\!\cdots\!34$$$$p^{22} T^{66} +$$$$33\!\cdots\!96$$$$p^{23} T^{67} +$$$$32\!\cdots\!03$$$$p^{24} T^{68} +$$$$30\!\cdots\!59$$$$p^{25} T^{69} +$$$$28\!\cdots\!79$$$$p^{26} T^{70} +$$$$25\!\cdots\!27$$$$p^{27} T^{71} +$$$$21\!\cdots\!83$$$$p^{28} T^{72} + 18200099284451033495 p^{29} T^{73} + 34600523118082039 p^{31} T^{74} + 114297937396905555 p^{31} T^{75} + 201812637620028 p^{33} T^{76} + 610852036948918 p^{33} T^{77} + 42591813101713 p^{34} T^{78} + 2698739936154 p^{35} T^{79} + 170189860166 p^{36} T^{80} + 9449040322 p^{37} T^{81} + 527429182 p^{38} T^{82} + 24529122 p^{39} T^{83} + 1173233 p^{40} T^{84} + 41836 p^{41} T^{85} + 1626 p^{42} T^{86} + 35 p^{43} T^{87} + p^{44} T^{88}$$
47 $$1 + 45 T + 2156 T^{2} + 64758 T^{3} + 1891117 T^{4} + 44017299 T^{5} + 982670399 T^{6} + 19012756743 T^{7} + 352887480769 T^{8} + 5909412432184 T^{9} + 95241848578296 T^{10} + 1417019068573525 T^{11} + 20368565045332523 T^{12} + 274243109673249124 T^{13} + 3580588927460164347 T^{14} + 44229499648602413917 T^{15} +$$$$53\!\cdots\!08$$$$T^{16} +$$$$60\!\cdots\!50$$$$T^{17} +$$$$68\!\cdots\!71$$$$T^{18} +$$$$72\!\cdots\!99$$$$T^{19} +$$$$76\!\cdots\!00$$$$T^{20} +$$$$77\!\cdots\!40$$$$T^{21} +$$$$76\!\cdots\!34$$$$T^{22} +$$$$73\!\cdots\!45$$$$T^{23} +$$$$68\!\cdots\!66$$$$T^{24} +$$$$62\!\cdots\!96$$$$T^{25} +$$$$56\!\cdots\!54$$$$T^{26} +$$$$49\!\cdots\!72$$$$T^{27} +$$$$42\!\cdots\!55$$$$T^{28} +$$$$35\!\cdots\!91$$$$T^{29} +$$$$29\!\cdots\!73$$$$T^{30} +$$$$24\!\cdots\!10$$$$T^{31} +$$$$19\!\cdots\!55$$$$T^{32} +$$$$15\!\cdots\!74$$$$T^{33} +$$$$12\!\cdots\!30$$$$T^{34} +$$$$91\!\cdots\!88$$$$T^{35} +$$$$69\!\cdots\!91$$$$T^{36} +$$$$51\!\cdots\!32$$$$T^{37} +$$$$38\!\cdots\!88$$$$T^{38} +$$$$27\!\cdots\!99$$$$T^{39} +$$$$20\!\cdots\!94$$$$T^{40} +$$$$14\!\cdots\!53$$$$T^{41} +$$$$10\!\cdots\!39$$$$T^{42} +$$$$70\!\cdots\!75$$$$T^{43} +$$$$48\!\cdots\!27$$$$T^{44} +$$$$70\!\cdots\!75$$$$p T^{45} +$$$$10\!\cdots\!39$$$$p^{2} T^{46} +$$$$14\!\cdots\!53$$$$p^{3} T^{47} +$$$$20\!\cdots\!94$$$$p^{4} T^{48} +$$$$27\!\cdots\!99$$$$p^{5} T^{49} +$$$$38\!\cdots\!88$$$$p^{6} T^{50} +$$$$51\!\cdots\!32$$$$p^{7} T^{51} +$$$$69\!\cdots\!91$$$$p^{8} T^{52} +$$$$91\!\cdots\!88$$$$p^{9} T^{53} +$$$$12\!\cdots\!30$$$$p^{10} T^{54} +$$$$15\!\cdots\!74$$$$p^{11} T^{55} +$$$$19\!\cdots\!55$$$$p^{12} T^{56} +$$$$24\!\cdots\!10$$$$p^{13} T^{57} +$$$$29\!\cdots\!73$$$$p^{14} T^{58} +$$$$35\!\cdots\!91$$$$p^{15} T^{59} +$$$$42\!\cdots\!55$$$$p^{16} T^{60} +$$$$49\!\cdots\!72$$$$p^{17} T^{61} +$$$$56\!\cdots\!54$$$$p^{18} T^{62} +$$$$62\!\cdots\!96$$$$p^{19} T^{63} +$$$$68\!\cdots\!66$$$$p^{20} T^{64} +$$$$73\!\cdots\!45$$$$p^{21} T^{65} +$$$$76\!\cdots\!34$$$$p^{22} T^{66} +$$$$77\!\cdots\!40$$$$p^{23} T^{67} +$$$$76\!\cdots\!00$$$$p^{24} T^{68} +$$$$72\!\cdots\!99$$$$p^{25} T^{69} +$$$$68\!\cdots\!71$$$$p^{26} T^{70} +$$$$60\!\cdots\!50$$$$p^{27} T^{71} +$$$$53\!\cdots\!08$$$$p^{28} T^{72} + 44229499648602413917 p^{29} T^{73} + 3580588927460164347 p^{30} T^{74} + 274243109673249124 p^{31} T^{75} + 20368565045332523 p^{32} T^{76} + 1417019068573525 p^{33} T^{77} + 95241848578296 p^{34} T^{78} + 5909412432184 p^{35} T^{79} + 352887480769 p^{36} T^{80} + 19012756743 p^{37} T^{81} + 982670399 p^{38} T^{82} + 44017299 p^{39} T^{83} + 1891117 p^{40} T^{84} + 64758 p^{41} T^{85} + 2156 p^{42} T^{86} + 45 p^{43} T^{87} + p^{44} T^{88}$$
53 $$1 - 18 T + 1480 T^{2} - 25087 T^{3} + 1093363 T^{4} - 17477272 T^{5} + 536793258 T^{6} - 8108482422 T^{7} + 196816208362 T^{8} - 2816162666772 T^{9} + 57435198986459 T^{10} - 780413546500014 T^{11} + 13886077376454952 T^{12} - 179617131146250173 T^{13} + 2859135861624590352 T^{14} - 35289737213642734580 T^{15} +$$$$51\!\cdots\!07$$$$T^{16} -$$$$60\!\cdots\!38$$$$T^{17} +$$$$80\!\cdots\!62$$$$T^{18} -$$$$91\!\cdots\!93$$$$T^{19} +$$$$11\!\cdots\!67$$$$T^{20} -$$$$12\!\cdots\!42$$$$T^{21} +$$$$14\!\cdots\!44$$$$T^{22} -$$$$15\!\cdots\!33$$$$T^{23} +$$$$16\!\cdots\!01$$$$T^{24} -$$$$16\!\cdots\!55$$$$T^{25} +$$$$17\!\cdots\!85$$$$T^{26} -$$$$17\!\cdots\!29$$$$T^{27} +$$$$17\!\cdots\!97$$$$T^{28} -$$$$15\!\cdots\!82$$$$T^{29} +$$$$15\!\cdots\!51$$$$T^{30} -$$$$13\!\cdots\!49$$$$T^{31} +$$$$12\!\cdots\!37$$$$T^{32} -$$$$11\!\cdots\!69$$$$T^{33} +$$$$97\!\cdots\!79$$$$T^{34} -$$$$82\!\cdots\!38$$$$T^{35} +$$$$69\!\cdots\!52$$$$T^{36} -$$$$57\!\cdots\!69$$$$T^{37} +$$$$46\!\cdots\!80$$$$T^{38} -$$$$37\!\cdots\!81$$$$T^{39} +$$$$29\!\cdots\!53$$$$T^{40} -$$$$22\!\cdots\!47$$$$T^{41} +$$$$17\!\cdots\!09$$$$T^{42} -$$$$12\!\cdots\!85$$$$T^{43} +$$$$93\!\cdots\!74$$$$T^{44} -$$$$12\!\cdots\!85$$$$p T^{45} +$$$$17\!\cdots\!09$$$$p^{2} T^{46} -$$$$22\!\cdots\!47$$$$p^{3} T^{47} +$$$$29\!\cdots\!53$$$$p^{4} T^{48} -$$$$37\!\cdots\!81$$$$p^{5} T^{49} +$$$$46\!\cdots\!80$$$$p^{6} T^{50} -$$$$57\!\cdots\!69$$$$p^{7} T^{51} +$$$$69\!\cdots\!52$$$$p^{8} T^{52} -$$$$82\!\cdots\!38$$$$p^{9} T^{53} +$$$$97\!\cdots\!79$$$$p^{10} T^{54} -$$$$11\!\cdots\!69$$$$p^{11} T^{55} +$$$$12\!\cdots\!37$$$$p^{12} T^{56} -$$$$13\!\cdots\!49$$$$p^{13} T^{57} +$$$$15\!\cdots\!51$$$$p^{14} T^{58} -$$$$15\!\cdots\!82$$$$p^{15} T^{59} +$$$$17\!\cdots\!97$$$$p^{16} T^{60} -$$$$17\!\cdots\!29$$$$p^{17} T^{61} +$$$$17\!\cdots\!85$$$$p^{18} T^{62} -$$$$16\!\cdots\!55$$$$p^{19} T^{63} +$$$$16\!\cdots\!01$$$$p^{20} T^{64} -$$$$15\!\cdots\!33$$$$p^{21} T^{65} +$$$$14\!\cdots\!44$$$$p^{22} T^{66} -$$$$12\!\cdots\!42$$$$p^{23} T^{67} +$$$$11\!\cdots\!67$$$$p^{24} T^{68} -$$$$91\!\cdots\!93$$$$p^{25} T^{69} +$$$$80\!\cdots\!62$$$$p^{26} T^{70} -$$$$60\!\cdots\!38$$$$p^{27} T^{71} +$$$$51\!\cdots\!07$$$$p^{28} T^{72} - 35289737213642734580 p^{29} T^{73} + 2859135861624590352 p^{30} T^{74} - 179617131146250173 p^{31} T^{75} + 13886077376454952 p^{32} T^{76} - 780413546500014 p^{33} T^{77} + 57435198986459 p^{34} T^{78} - 2816162666772 p^{35} T^{79} + 196816208362 p^{36} T^{80} - 8108482422 p^{37} T^{81} + 536793258 p^{38} T^{82} - 17477272 p^{39} T^{83} + 1093363 p^{40} T^{84} - 25087 p^{41} T^{85} + 1480 p^{42} T^{86} - 18 p^{43} T^{87} + p^{44} T^{88}$$
59 $$1 + 57 T + 2909 T^{2} + 104136 T^{3} + 3355738 T^{4} + 91798602 T^{5} + 2308581354 T^{6} + 52399412451 T^{7} + 1111021568356 T^{8} + 21868410596175 T^{9} + 406703996078666 T^{10} + 7132520814370032 T^{11} + 119163819801113018 T^{12} + 1895913550454332010 T^{13} + 28911278981144677574 T^{14} +$$$$42\!\cdots\!39$$$$T^{15} +$$$$59\!\cdots\!26$$$$T^{16} +$$$$80\!\cdots\!32$$$$T^{17} +$$$$10\!\cdots\!09$$$$T^{18} +$$$$13\!\cdots\!99$$$$T^{19} +$$$$16\!\cdots\!67$$$$T^{20} +$$$$19\!\cdots\!35$$$$T^{21} +$$$$22\!\cdots\!17$$$$T^{22} +$$$$25\!\cdots\!31$$$$T^{23} +$$$$27\!\cdots\!33$$$$T^{24} +$$$$29\!\cdots\!28$$$$T^{25} +$$$$30\!\cdots\!05$$$$T^{26} +$$$$30\!\cdots\!99$$$$T^{27} +$$$$30\!\cdots\!52$$$$T^{28} +$$$$29\!\cdots\!00$$$$T^{29} +$$$$27\!\cdots\!89$$$$T^{30} +$$$$25\!\cdots\!23$$$$T^{31} +$$$$22\!\cdots\!21$$$$T^{32} +$$$$20\!\cdots\!72$$$$T^{33} +$$$$17\!\cdots\!96$$$$T^{34} +$$$$14\!\cdots\!30$$$$T^{35} +$$$$12\!\cdots\!83$$$$T^{36} +$$$$10\!\cdots\!18$$$$T^{37} +$$$$82\!\cdots\!40$$$$T^{38} +$$$$66\!\cdots\!27$$$$T^{39} +$$$$52\!\cdots\!17$$$$T^{40} +$$$$40\!\cdots\!51$$$$T^{41} +$$$$31\!\cdots\!73$$$$T^{42} +$$$$24\!\cdots\!17$$$$T^{43} +$$$$19\!\cdots\!12$$$$T^{44} +$$$$24\!\cdots\!17$$$$p T^{45} +$$$$31\!\cdots\!73$$$$p^{2} T^{46} +$$$$40\!\cdots\!51$$$$p^{3} T^{47} +$$$$52\!\cdots\!17$$$$p^{4} T^{48} +$$$$66\!\cdots\!27$$$$p^{5} T^{49} +$$$$82\!\cdots\!40$$$$p^{6} T^{50} +$$$$10\!\cdots\!18$$$$p^{7} T^{51} +$$$$12\!\cdots\!83$$$$p^{8} T^{52} +$$$$14\!\cdots\!30$$$$p^{9} T^{53} +$$$$17\!\cdots\!96$$$$p^{10} T^{54} +$$$$20\!\cdots\!72$$$$p^{11} T^{55} +$$$$22\!\cdots\!21$$$$p^{12} T^{56} +$$$$25\!\cdots\!23$$$$p^{13} T^{57} +$$$$27\!\cdots\!89$$$$p^{14} T^{58} +$$$$29\!\cdots\!00$$$$p^{15} T^{59} +$$$$30\!\cdots\!52$$$$p^{16} T^{60} +$$$$30\!\cdots\!99$$$$p^{17} T^{61} +$$$$30\!\cdots\!05$$$$p^{18} T^{62} +$$$$29\!\cdots\!28$$$$p^{19} T^{63} +$$$$27\!\cdots\!33$$$$p^{20} T^{64} +$$$$25\!\cdots\!31$$$$p^{21} T^{65} +$$$$22\!\cdots\!17$$$$p^{22} T^{66} +$$$$19\!\cdots\!35$$$$p^{23} T^{67} +$$$$16\!\cdots\!67$$$$p^{24} T^{68} +$$$$13\!\cdots\!99$$$$p^{25} T^{69} +$$$$10\!\cdots\!09$$$$p^{26} T^{70} +$$$$80\!\cdots\!32$$$$p^{27} T^{71} +$$$$59\!\cdots\!26$$$$p^{28} T^{72} +$$$$42\!\cdots\!39$$$$p^{29} T^{73} + 28911278981144677574 p^{30} T^{74} + 1895913550454332010 p^{31} T^{75} + 119163819801113018 p^{32} T^{76} + 7132520814370032 p^{33} T^{77} + 406703996078666 p^{34} T^{78} + 21868410596175 p^{35} T^{79} + 1111021568356 p^{36} T^{80} + 52399412451 p^{37} T^{81} + 2308581354 p^{38} T^{82} + 91798602 p^{39} T^{83} + 3355738 p^{40} T^{84} + 104136 p^{41} T^{85} + 2909 p^{42} T^{86} + 57 p^{43} T^{87} + p^{44} T^{88}$$
61 $$1 - 27 T + 1747 T^{2} - 40682 T^{3} + 1481839 T^{4} - 30558873 T^{5} + 820313700 T^{6} - 15267380611 T^{7} + 335182371320 T^{8} - 5710248843964 T^{9} + 108219842167179 T^{10} - 1706099320621174 T^{11} + 28832724259743162 T^{12} - 424273261188319399 T^{13} + 6531572240216626486 T^{14} - 90333570194751531141 T^{15} +$$$$12\!\cdots\!88$$$$T^{16} -$$$$16\!\cdots\!67$$$$T^{17} +$$$$22\!\cdots\!54$$$$T^{18} -$$$$27\!\cdots\!90$$$$T^{19} +$$$$34\!\cdots\!67$$$$T^{20} -$$$$41\!\cdots\!33$$$$T^{21} +$$$$48\!\cdots\!41$$$$T^{22} -$$$$55\!\cdots\!32$$$$T^{23} +$$$$62\!\cdots\!35$$$$T^{24} -$$$$68\!\cdots\!18$$$$T^{25} +$$$$73\!\cdots\!18$$$$T^{26} -$$$$77\!\cdots\!42$$$$T^{27} +$$$$80\!\cdots\!19$$$$T^{28} -$$$$80\!\cdots\!11$$$$T^{29} +$$$$80\!\cdots\!89$$$$T^{30} -$$$$77\!\cdots\!43$$$$T^{31} +$$$$75\!\cdots\!89$$$$T^{32} -$$$$70\!\cdots\!88$$$$T^{33} +$$$$65\!\cdots\!15$$$$T^{34} -$$$$59\!\cdots\!62$$$$T^{35} +$$$$53\!\cdots\!21$$$$T^{36} -$$$$46\!\cdots\!82$$$$T^{37} +$$$$40\!\cdots\!12$$$$T^{38} -$$$$34\!\cdots\!19$$$$T^{39} +$$$$28\!\cdots\!63$$$$T^{40} -$$$$23\!\cdots\!09$$$$T^{41} +$$$$19\!\cdots\!02$$$$T^{42} -$$$$15\!\cdots\!57$$$$T^{43} +$$$$12\!\cdots\!58$$$$T^{44} -$$$$15\!\cdots\!57$$$$p T^{45} +$$$$19\!\cdots\!02$$$$p^{2} T^{46} -$$$$23\!\cdots\!09$$$$p^{3} T^{47} +$$$$28\!\cdots\!63$$$$p^{4} T^{48} -$$$$34\!\cdots\!19$$$$p^{5} T^{49} +$$$$40\!\cdots\!12$$$$p^{6} T^{50} -$$$$46\!\cdots\!82$$$$p^{7} T^{51} +$$$$53\!\cdots\!21$$$$p^{8} T^{52} -$$$$59\!\cdots\!62$$$$p^{9} T^{53} +$$$$65\!\cdots\!15$$$$p^{10} T^{54} -$$$$70\!\cdots\!88$$$$p^{11} T^{55} +$$$$75\!\cdots\!89$$$$p^{12} T^{56} -$$$$77\!\cdots\!43$$$$p^{13} T^{57} +$$$$80\!\cdots\!89$$$$p^{14} T^{58} -$$$$80\!\cdots\!11$$$$p^{15} T^{59} +$$$$80\!\cdots\!19$$$$p^{16} T^{60} -$$$$77\!\cdots\!42$$$$p^{17} T^{61} +$$$$73\!\cdots\!18$$$$p^{18} T^{62} -$$$$68\!\cdots\!18$$$$p^{19} T^{63} +$$$$62\!\cdots\!35$$$$p^{20} T^{64} -$$$$55\!\cdots\!32$$$$p^{21} T^{65} +$$$$48\!\cdots\!41$$$$p^{22} T^{66} -$$$$41\!\cdots\!33$$$$p^{23} T^{67} +$$$$34\!\cdots\!67$$$$p^{24} T^{68} -$$$$27\!\cdots\!90$$$$p^{25} T^{69} +$$$$22\!\cdots\!54$$$$p^{26} T^{70} -$$$$16\!\cdots\!67$$$$p^{27} T^{71} +$$$$12\!\cdots\!88$$$$p^{28} T^{72} - 90333570194751531141 p^{29} T^{73} + 6531572240216626486 p^{30} T^{74} - 424273261188319399 p^{31} T^{75} + 28832724259743162 p^{32} T^{76} - 1706099320621174 p^{33} T^{77} + 108219842167179 p^{34} T^{78} - 5710248843964 p^{35} T^{79} + 335182371320 p^{36} T^{80} - 15267380611 p^{37} T^{81} + 820313700 p^{38} T^{82} - 30558873 p^{39} T^{83} + 1481839 p^{40} T^{84} - 40682 p^{41} T^{85} + 1747 p^{42} T^{86} - 27 p^{43} T^{87} + p^{44} T^{88}$$
67 $$1 + 56 T + 2937 T^{2} + 102745 T^{3} + 3325011 T^{4} + 88362831 T^{5} + 2198854880 T^{6} + 48240626948 T^{7} + 1002304061969 T^{8} + 18998056686166 T^{9} + 344078973819592 T^{10} + 5797497883976470 T^{11} + 93991663004923139 T^{12} + 1435532466600004700 T^{13} + 21214417233495560878 T^{14} +$$$$29\!\cdots\!69$$$$T^{15} +$$$$40\!\cdots\!86$$$$T^{16} +$$$$53\!\cdots\!70$$$$T^{17} +$$$$67\!\cdots\!52$$$$T^{18} +$$$$82\!\cdots\!15$$$$T^{19} +$$$$99\!\cdots\!53$$$$T^{20} +$$$$11\!\cdots\!54$$$$T^{21} +$$$$13\!\cdots\!80$$$$T^{22} +$$$$14\!\cdots\!21$$$$T^{23} +$$$$15\!\cdots\!66$$$$T^{24} +$$$$16\!\cdots\!20$$$$T^{25} +$$$$16\!\cdots\!55$$$$T^{26} +$$$$17\!\cdots\!87$$$$T^{27} +$$$$17\!\cdots\!00$$$$T^{28} +$$$$16\!\cdots\!37$$$$T^{29} +$$$$16\!\cdots\!41$$$$T^{30} +$$$$15\!\cdots\!61$$$$T^{31} +$$$$14\!\cdots\!61$$$$T^{32} +$$$$13\!\cdots\!50$$$$T^{33} +$$$$12\!\cdots\!69$$$$T^{34} +$$$$10\!\cdots\!19$$$$T^{35} +$$$$98\!\cdots\!68$$$$T^{36} +$$$$86\!\cdots\!22$$$$T^{37} +$$$$75\!\cdots\!82$$$$T^{38} +$$$$65\!\cdots\!71$$$$T^{39} +$$$$56\!\cdots\!65$$$$T^{40} +$$$$47\!\cdots\!44$$$$T^{41} +$$$$40\!\cdots\!38$$$$T^{42} +$$$$33\!\cdots\!84$$$$T^{43} +$$$$27\!\cdots\!42$$$$T^{44} +$$$$33\!\cdots\!84$$$$p T^{45} +$$$$40\!\cdots\!38$$$$p^{2} T^{46} +$$$$47\!\cdots\!44$$$$p^{3} T^{47} +$$$$56\!\cdots\!65$$$$p^{4} T^{48} +$$$$65\!\cdots\!71$$$$p^{5} T^{49} +$$$$75\!\cdots\!82$$$$p^{6} T^{50} +$$$$86\!\cdots\!22$$$$p^{7} T^{51} +$$$$98\!\cdots\!68$$$$p^{8} T^{52} +$$$$10\!\cdots\!19$$$$p^{9} T^{53} +$$$$12\!\cdots\!69$$$$p^{10} T^{54} +$$$$13\!\cdots\!50$$$$p^{11} T^{55} +$$$$14\!\cdots\!61$$$$p^{12} T^{56} +$$$$15\!\cdots\!61$$$$p^{13} T^{57} +$$$$16\!\cdots\!41$$$$p^{14} T^{58} +$$$$16\!\cdots\!37$$$$p^{15} T^{59} +$$$$17\!\cdots\!00$$$$p^{16} T^{60} +$$$$17\!\cdots\!87$$$$p^{17} T^{61} +$$$$16\!\cdots\!55$$$$p^{18} T^{62} +$$$$16\!\cdots\!20$$$$p^{19} T^{63} +$$$$15\!\cdots\!66$$$$p^{20} T^{64} +$$$$14\!\cdots\!21$$$$p^{21} T^{65} +$$$$13\!\cdots\!80$$$$p^{22} T^{66} +$$$$11\!\cdots\!54$$$$p^{23} T^{67} +$$$$99\!\cdots\!53$$$$p^{24} T^{68} +$$$$82\!\cdots\!15$$$$p^{25} T^{69} +$$$$67\!\cdots\!52$$$$p^{26} T^{70} +$$$$53\!\cdots\!70$$$$p^{27} T^{71} +$$$$40\!\cdots\!86$$$$p^{28} T^{72} +$$$$29\!\cdots\!69$$$$p^{29} T^{73} + 21214417233495560878 p^{30} T^{74} + 1435532466600004700 p^{31} T^{75} + 93991663004923139 p^{32} T^{76} + 5797497883976470 p^{33} T^{77} + 344078973819592 p^{34} T^{78} + 18998056686166 p^{35} T^{79} + 1002304061969 p^{36} T^{80} + 48240626948 p^{37} T^{81} + 2198854880 p^{38} T^{82} + 88362831 p^{39} T^{83} + 3325011 p^{40} T^{84} + 102745 p^{41} T^{85} + 2937 p^{42} T^{86} + 56 p^{43} T^{87} + p^{44} T^{88}$$
71 $$1 + 52 T + 2658 T^{2} + 89408 T^{3} + 2840963 T^{4} + 74021251 T^{5} + 1828074132 T^{6} + 39937956493 T^{7} + 833167710685 T^{8} + 15946278826351 T^{9} + 293380173806868 T^{10} + 5055927378528906 T^{11} + 84209650807128427 T^{12} + 1331176650108922477 T^{13} + 20425482080110536038 T^{14} +$$$$30\!\cdots\!52$$$$T^{15} +$$$$42\!\cdots\!93$$$$T^{16} +$$$$59\!\cdots\!46$$$$T^{17} +$$$$79\!\cdots\!30$$$$T^{18} +$$$$10\!\cdots\!76$$$$T^{19} +$$$$13\!\cdots\!24$$$$T^{20} +$$$$16\!\cdots\!95$$$$T^{21} +$$$$20\!\cdots\!65$$$$T^{22} +$$$$23\!\cdots\!84$$$$T^{23} +$$$$27\!\cdots\!64$$$$T^{24} +$$$$31\!\cdots\!60$$$$T^{25} +$$$$35\!\cdots\!49$$$$T^{26} +$$$$38\!\cdots\!50$$$$T^{27} +$$$$42\!\cdots\!53$$$$T^{28} +$$$$44\!\cdots\!77$$$$T^{29} +$$$$46\!\cdots\!21$$$$T^{30} +$$$$47\!\cdots\!37$$$$T^{31} +$$$$47\!\cdots\!30$$$$T^{32} +$$$$47\!\cdots\!15$$$$T^{33} +$$$$46\!\cdots\!70$$$$T^{34} +$$$$44\!\cdots\!44$$$$T^{35} +$$$$42\!\cdots\!48$$$$T^{36} +$$$$39\!\cdots\!77$$$$T^{37} +$$$$36\!\cdots\!07$$$$T^{38} +$$$$33\!\cdots\!56$$$$T^{39} +$$$$42\!\cdots\!09$$$$p T^{40} +$$$$26\!\cdots\!92$$$$T^{41} +$$$$23\!\cdots\!14$$$$T^{42} +$$$$19\!\cdots\!99$$$$T^{43} +$$$$16\!\cdots\!98$$$$T^{44} +$$$$19\!\cdots\!99$$$$p T^{45} +$$$$23\!\cdots\!14$$$$p^{2} T^{46} +$$$$26\!\cdots\!92$$$$p^{3} T^{47} +$$$$42\!\cdots\!09$$$$p^{5} T^{48} +$$$$33\!\cdots\!56$$$$p^{5} T^{49} +$$$$36\!\cdots\!07$$$$p^{6} T^{50} +$$$$39\!\cdots\!77$$$$p^{7} T^{51} +$$$$42\!\cdots\!48$$$$p^{8} T^{52} +$$$$44\!\cdots\!44$$$$p^{9} T^{53} +$$$$46\!\cdots\!70$$$$p^{10} T^{54} +$$$$47\!\cdots\!15$$$$p^{11} T^{55} +$$$$47\!\cdots\!30$$$$p^{12} T^{56} +$$$$47\!\cdots\!37$$$$p^{13} T^{57} +$$$$46\!\cdots\!21$$$$p^{14} T^{58} +$$$$44\!\cdots\!77$$$$p^{15} T^{59} +$$$$42\!\cdots\!53$$$$p^{16} T^{60} +$$$$38\!\cdots\!50$$$$p^{17} T^{61} +$$$$35\!\cdots\!49$$$$p^{18} T^{62} +$$$$31\!\cdots\!60$$$$p^{19} T^{63} +$$$$27\!\cdots\!64$$$$p^{20} T^{64} +$$$$23\!\cdots\!84$$$$p^{21} T^{65} +$$$$20\!\cdots\!65$$$$p^{22} T^{66} +$$$$16\!\cdots\!95$$$$p^{23} T^{67} +$$$$13\!\cdots\!24$$$$p^{24} T^{68} +$$$$10\!\cdots\!76$$$$p^{25} T^{69} +$$$$79\!\cdots\!30$$$$p^{26} T^{70} +$$$$59\!\cdots\!46$$$$p^{27} T^{71} +$$$$42\!\cdots\!93$$$$p^{28} T^{72} +$$$$30\!\cdots\!52$$$$p^{29} T^{73} + 20425482080110536038 p^{30} T^{74} + 1331176650108922477 p^{31} T^{75} + 84209650807128427 p^{32} T^{76} + 5055927378528906 p^{33} T^{77} + 293380173806868 p^{34} T^{78} + 15946278826351 p^{35} T^{79} + 833167710685 p^{36} T^{80} + 39937956493 p^{37} T^{81} + 1828074132 p^{38} T^{82} + 74021251 p^{39} T^{83} + 2840963 p^{40} T^{84} + 89408 p^{41} T^{85} + 2658 p^{42} T^{86} + 52 p^{43} T^{87} + p^{44} T^{88}$$
73 $$1 + 68 T + 3665 T^{2} + 146005 T^{3} + 5051564 T^{4} + 151504238 T^{5} + 4121875024 T^{6} + 102272444487 T^{7} + 2358190204907 T^{8} + 50793536998798 T^{9} + 1031956447865372 T^{10} + 19857736084830042 T^{11} + 364024090755805050 T^{12} + 6377404291946467890 T^{13} +$$$$10\!\cdots\!75$$$$T^{14} +$$$$17\!\cdots\!18$$$$T^{15} +$$$$27\!\cdots\!26$$$$T^{16} +$$$$40\!\cdots\!86$$$$T^{17} +$$$$59\!\cdots\!28$$$$T^{18} +$$$$83\!\cdots\!97$$$$T^{19} +$$$$11\!\cdots\!07$$$$T^{20} +$$$$15\!\cdots\!69$$$$T^{21} +$$$$19\!\cdots\!98$$$$T^{22} +$$$$25\!\cdots\!51$$$$T^{23} +$$$$31\!\cdots\!55$$$$T^{24} +$$$$37\!\cdots\!13$$$$T^{25} +$$$$44\!\cdots\!24$$$$T^{26} +$$$$51\!\cdots\!99$$$$T^{27} +$$$$58\!\cdots\!52$$$$T^{28} +$$$$64\!\cdots\!46$$$$T^{29} +$$$$70\!\cdots\!35$$$$T^{30} +$$$$74\!\cdots\!81$$$$T^{31} +$$$$77\!\cdots\!20$$$$T^{32} +$$$$79\!\cdots\!31$$$$T^{33} +$$$$80\!\cdots\!06$$$$T^{34} +$$$$79\!\cdots\!39$$$$T^{35} +$$$$77\!\cdots\!17$$$$T^{36} +$$$$74\!\cdots\!92$$$$T^{37} +$$$$70\!\cdots\!95$$$$T^{38} +$$$$65\!\cdots\!37$$$$T^{39} +$$$$59\!\cdots\!47$$$$T^{40} +$$$$53\!\cdots\!74$$$$T^{41} +$$$$47\!\cdots\!22$$$$T^{42} +$$$$41\!\cdots\!43$$$$T^{43} +$$$$35\!\cdots\!44$$$$T^{44} +$$$$41\!\cdots\!43$$$$p T^{45} +$$$$47\!\cdots\!22$$$$p^{2} T^{46} +$$$$53\!\cdots\!74$$$$p^{3} T^{47} +$$$$59\!\cdots\!47$$$$p^{4} T^{48} +$$$$65\!\cdots\!37$$$$p^{5} T^{49} +$$$$70\!\cdots\!95$$$$p^{6} T^{50} +$$$$74\!\cdots\!92$$$$p^{7} T^{51} +$$$$77\!\cdots\!17$$$$p^{8} T^{52} +$$$$79\!\cdots\!39$$$$p^{9} T^{53} +$$$$80\!\cdots\!06$$$$p^{10} T^{54} +$$$$79\!\cdots\!31$$$$p^{11} T^{55} +$$$$77\!\cdots\!20$$$$p^{12} T^{56} +$$$$74\!\cdots\!81$$$$p^{13} T^{57} +$$$$70\!\cdots\!35$$$$p^{14} T^{58} +$$$$64\!\cdots\!46$$$$p^{15} T^{59} +$$$$58\!\cdots\!52$$$$p^{16} T^{60} +$$$$51\!\cdots\!99$$$$p^{17} T^{61} +$$$$44\!\cdots\!24$$$$p^{18} T^{62} +$$$$37\!\cdots\!13$$$$p^{19} T^{63} +$$$$31\!\cdots\!55$$$$p^{20} T^{64} +$$$$25\!\cdots\!51$$$$p^{21} T^{65} +$$$$19\!\cdots\!98$$$$p^{22} T^{66} +$$$$15\!\cdots\!69$$$$p^{23} T^{67} +$$$$11\!\cdots\!07$$$$p^{24} T^{68} +$$$$83\!\cdots\!97$$$$p^{25} T^{69} +$$$$59\!\cdots\!28$$$$p^{26} T^{70} +$$$$40\!\cdots\!86$$$$p^{27} T^{71} +$$$$27\!\cdots\!26$$$$p^{28} T^{72} +$$$$17\!\cdots\!18$$$$p^{29} T^{73} +$$$$10\!\cdots\!75$$$$p^{30} T^{74} + 6377404291946467890 p^{31} T^{75} + 364024090755805050 p^{32} T^{76} + 19857736084830042 p^{33} T^{77} + 1031956447865372 p^{34} T^{78} + 50793536998798 p^{35} T^{79} + 2358190204907 p^{36} T^{80} + 102272444487 p^{37} T^{81} + 4121875024 p^{38} T^{82} + 151504238 p^{39} T^{83} + 5051564 p^{40} T^{84} + 146005 p^{41} T^{85} + 3665 p^{42} T^{86} + 68 p^{43} T^{87} + p^{44} T^{88}$$
79 $$1 + 55 T + 2953 T^{2} + 104486 T^{3} + 3478841 T^{4} + 94822736 T^{5} + 2442812138 T^{6} + 55496122006 T^{7} + 1201165058098 T^{8} + 23760113853853 T^{9} + 451014970367044 T^{10} + 7986349553134170 T^{11} + 136523714232237598 T^{12} + 2206063055150718867 T^{13} + 34585002672755430782 T^{14} +$$$$51\!\cdots\!87$$$$T^{15} +$$$$75\!\cdots\!64$$$$T^{16} +$$$$10\!\cdots\!37$$$$T^{17} +$$$$14\!\cdots\!85$$$$T^{18} +$$$$24\!\cdots\!39$$$$p T^{19} +$$$$24\!\cdots\!77$$$$T^{20} +$$$$31\!\cdots\!65$$$$T^{21} +$$$$38\!\cdots\!15$$$$T^{22} +$$$$45\!\cdots\!84$$$$T^{23} +$$$$54\!\cdots\!96$$$$T^{24} +$$$$62\!\cdots\!17$$$$T^{25} +$$$$71\!\cdots\!65$$$$T^{26} +$$$$79\!\cdots\!93$$$$T^{27} +$$$$87\!\cdots\!95$$$$T^{28} +$$$$94\!\cdots\!50$$$$T^{29} +$$$$10\!\cdots\!15$$$$T^{30} +$$$$10\!\cdots\!21$$$$T^{31} +$$$$10\!\cdots\!55$$$$T^{32} +$$$$11\!\cdots\!06$$$$T^{33} +$$$$11\!\cdots\!92$$$$T^{34} +$$$$11\!\cdots\!85$$$$T^{35} +$$$$10\!\cdots\!56$$$$T^{36} +$$$$10\!\cdots\!53$$$$T^{37} +$$$$10\!\cdots\!01$$$$T^{38} +$$$$95\!\cdots\!74$$$$T^{39} +$$$$89\!\cdots\!06$$$$T^{40} +$$$$82\!\cdots\!04$$$$T^{41} +$$$$75\!\cdots\!06$$$$T^{42} +$$$$86\!\cdots\!28$$$$p T^{43} +$$$$61\!\cdots\!54$$$$T^{44} +$$$$86\!\cdots\!28$$$$p^{2} T^{45} +$$$$75\!\cdots\!06$$$$p^{2} T^{46} +$$$$82\!\cdots\!04$$$$p^{3} T^{47} +$$$$89\!\cdots\!06$$$$p^{4} T^{48} +$$$$95\!\cdots\!74$$$$p^{5} T^{49} +$$$$10\!\cdots\!01$$$$p^{6} T^{50} +$$$$10\!\cdots\!53$$$$p^{7} T^{51} +$$$$10\!\cdots\!56$$$$p^{8} T^{52} +$$$$11\!\cdots\!85$$$$p^{9} T^{53} +$$$$11\!\cdots\!92$$$$p^{10} T^{54} +$$$$11\!\cdots\!06$$$$p^{11} T^{55} +$$$$10\!\cdots\!55$$$$p^{12} T^{56} +$$$$10\!\cdots\!21$$$$p^{13} T^{57} +$$$$10\!\cdots\!15$$$$p^{14} T^{58} +$$$$94\!\cdots\!50$$$$p^{15} T^{59} +$$$$87\!\cdots\!95$$$$p^{16} T^{60} +$$$$79\!\cdots\!93$$$$p^{17} T^{61} +$$$$71\!\cdots\!65$$$$p^{18} T^{62} +$$$$62\!\cdots\!17$$$$p^{19} T^{63} +$$$$54\!\cdots\!96$$$$p^{20} T^{64} +$$$$45\!\cdots\!84$$$$p^{21} T^{65} +$$$$38\!\cdots\!15$$$$p^{22} T^{66} +$$$$31\!\cdots\!65$$$$p^{23} T^{67} +$$$$24\!\cdots\!77$$$$p^{24} T^{68} +$$$$24\!\cdots\!39$$$$p^{26} T^{69} +$$$$14\!\cdots\!85$$$$p^{26} T^{70} +$$$$10\!\cdots\!37$$$$p^{27} T^{71} +$$$$75\!\cdots\!64$$$$p^{28} T^{72} +$$$$51\!\cdots\!87$$$$p^{29} T^{73} + 34585002672755430782 p^{30} T^{74} + 2206063055150718867 p^{31} T^{75} + 136523714232237598 p^{32} T^{76} + 7986349553134170 p^{33} T^{77} + 451014970367044 p^{34} T^{78} + 23760113853853 p^{35} T^{79} + 1201165058098 p^{36} T^{80} + 55496122006 p^{37} T^{81} + 2442812138 p^{38} T^{82} + 94822736 p^{39} T^{83} + 3478841 p^{40} T^{84} + 104486 p^{41} T^{85} + 2953 p^{42} T^{86} + 55 p^{43} T^{87} + p^{44} T^{88}$$
83 $$1 + 51 T + 2914 T^{2} + 100682 T^{3} + 3498213 T^{4} + 94522150 T^{5} + 2515048836 T^{6} + 56948418613 T^{7} + 1265367240040 T^{8} + 25006352940796 T^{9} + 485577423320332 T^{10} + 8602519265272774 T^{11} + 150163213991025044 T^{12} + 2430503682304765394 T^{13} + 38880794869212651105 T^{14} +$$$$58\!\cdots\!95$$$$T^{15} +$$$$86\!\cdots\!49$$$$T^{16} +$$$$12\!\cdots\!04$$$$T^{17} +$$$$16\!\cdots\!50$$$$T^{18} +$$$$22\!\cdots\!04$$$$T^{19} +$$$$29\!\cdots\!02$$$$T^{20} +$$$$37\!\cdots\!01$$$$T^{21} +$$$$47\!\cdots\!85$$$$T^{22} +$$$$57\!\cdots\!45$$$$T^{23} +$$$$68\!\cdots\!76$$$$T^{24} +$$$$79\!\cdots\!22$$$$T^{25} +$$$$92\!\cdots\!84$$$$T^{26} +$$$$10\!\cdots\!82$$$$T^{27} +$$$$11\!\cdots\!15$$$$T^{28} +$$$$12\!\cdots\!91$$$$T^{29} +$$$$13\!\cdots\!75$$$$T^{30} +$$$$14\!\cdots\!54$$$$T^{31} +$$$$15\!\cdots\!03$$$$T^{32} +$$$$15\!\cdots\!01$$$$T^{33} +$$$$15\!\cdots\!70$$$$T^{34} +$$$$15\!\cdots\!71$$$$T^{35} +$$$$15\!\cdots\!03$$$$T^{36} +$$$$15\!\cdots\!92$$$$T^{37} +$$$$15\!\cdots\!29$$$$T^{38} +$$$$14\!\cdots\!52$$$$T^{39} +$$$$13\!\cdots\!59$$$$T^{40} +$$$$12\!\cdots\!45$$$$T^{41} +$$$$12\!\cdots\!16$$$$T^{42} +$$$$10\!\cdots\!19$$$$T^{43} +$$$$10\!\cdots\!10$$$$T^{44} +$$$$10\!\cdots\!19$$$$p T^{45} +$$$$12\!\cdots\!16$$$$p^{2} T^{46} +$$$$12\!\cdots\!45$$$$p^{3} T^{47} +$$$$13\!\cdots\!59$$$$p^{4} T^{48} +$$$$14\!\cdots\!52$$$$p^{5} T^{49} +$$$$15\!\cdots\!29$$$$p^{6} T^{50} +$$$$15\!\cdots\!92$$$$p^{7} T^{51} +$$$$15\!\cdots\!03$$$$p^{8} T^{52} +$$$$15\!\cdots\!71$$$$p^{9} T^{53} +$$$$15\!\cdots\!70$$$$p^{10} T^{54} +$$$$15\!\cdots\!01$$$$p^{11} T^{55} +$$$$15\!\cdots\!03$$$$p^{12} T^{56} +$$$$14\!\cdots\!54$$$$p^{13} T^{57} +$$$$13\!\cdots\!75$$$$p^{14} T^{58} +$$$$12\!\cdots\!91$$$$p^{15} T^{59} +$$$$11\!\cdots\!15$$$$p^{16} T^{60} +$$$$10\!\cdots\!82$$$$p^{17} T^{61} +$$$$92\!\cdots\!84$$$$p^{18} T^{62} +$$$$79\!\cdots\!22$$$$p^{19} T^{63} +$$$$68\!\cdots\!76$$$$p^{20} T^{64} +$$$$57\!\cdots\!45$$$$p^{21} T^{65} +$$$$47\!\cdots\!85$$$$p^{22} T^{66} +$$$$37\!\cdots\!01$$$$p^{23} T^{67} +$$$$29\!\cdots\!02$$$$p^{24} T^{68} +$$$$22\!\cdots\!04$$$$p^{25} T^{69} +$$$$16\!\cdots\!50$$$$p^{26} T^{70} +$$$$12\!\cdots\!04$$$$p^{27} T^{71} +$$$$86\!\cdots\!49$$$$p^{28} T^{72} +$$$$58\!\cdots\!95$$$$p^{29} T^{73} + 38880794869212651105 p^{30} T^{74} + 2430503682304765394 p^{31} T^{75} + 150163213991025044 p^{32} T^{76} + 8602519265272774 p^{33} T^{77} + 485577423320332 p^{34} T^{78} + 25006352940796 p^{35} T^{79} + 1265367240040 p^{36} T^{80} + 56948418613 p^{37} T^{81} + 2515048836 p^{38} T^{82} + 94522150 p^{39} T^{83} + 3498213 p^{40} T^{84} + 100682 p^{41} T^{85} + 2914 p^{42} T^{86} + 51 p^{43} T^{87} + p^{44} T^{88}$$
89 $$1 + 21 T + 1888 T^{2} + 37551 T^{3} + 1799315 T^{4} + 33830358 T^{5} + 1151053135 T^{6} + 20464453312 T^{7} + 555021631313 T^{8} + 9345105080115 T^{9} + 214879531214828 T^{10} + 3433863156080202 T^{11} + 69508108424064762 T^{12} + 1056824922943738031 T^{13} + 19306783774357687228 T^{14} +$$$$27\!\cdots\!47$$$$T^{15} +$$$$46\!\cdots\!30$$$$T^{16} +$$$$65\!\cdots\!09$$$$T^{17} +$$$$10\!\cdots\!45$$$$T^{18} +$$$$13\!\cdots\!32$$$$T^{19} +$$$$19\!\cdots\!56$$$$T^{20} +$$$$25\!\cdots\!81$$$$T^{21} +$$$$35\!\cdots\!58$$$$T^{22} +$$$$43\!\cdots\!99$$$$T^{23} +$$$$56\!\cdots\!31$$$$T^{24} +$$$$67\!\cdots\!34$$$$T^{25} +$$$$85\!\cdots\!10$$$$T^{26} +$$$$97\!\cdots\!20$$$$T^{27} +$$$$11\!\cdots\!71$$$$T^{28} +$$$$13\!\cdots\!33$$$$T^{29} +$$$$15\!\cdots\!51$$$$T^{30} +$$$$16\!\cdots\!51$$$$T^{31} +$$$$18\!\cdots\!85$$$$T^{32} +$$$$19\!\cdots\!72$$$$T^{33} +$$$$23\!\cdots\!41$$$$p T^{34} +$$$$21\!\cdots\!79$$$$T^{35} +$$$$23\!\cdots\!42$$$$T^{36} +$$$$23\!\cdots\!20$$$$T^{37} +$$$$23\!\cdots\!63$$$$T^{38} +$$$$23\!\cdots\!58$$$$T^{39} +$$$$23\!\cdots\!34$$$$T^{40} +$$$$22\!\cdots\!22$$$$T^{41} +$$$$22\!\cdots\!84$$$$T^{42} +$$$$20\!\cdots\!11$$$$T^{43} +$$$$20\!\cdots\!22$$$$T^{44} +$$$$20\!\cdots\!11$$$$p T^{45} +$$$$22\!\cdots\!84$$$$p^{2} T^{46} +$$$$22\!\cdots\!22$$$$p^{3} T^{47} +$$$$23\!\cdots\!34$$$$p^{4} T^{48} +$$$$23\!\cdots\!58$$$$p^{5} T^{49} +$$$$23\!\cdots\!63$$$$p^{6} T^{50} +$$$$23\!\cdots\!20$$$$p^{7} T^{51} +$$$$23\!\cdots\!42$$$$p^{8} T^{52} +$$$$21\!\cdots\!79$$$$p^{9} T^{53} +$$$$23\!\cdots\!41$$$$p^{11} T^{54} +$$$$19\!\cdots\!72$$$$p^{11} T^{55} +$$$$18\!\cdots\!85$$$$p^{12} T^{56} +$$$$16\!\cdots\!51$$$$p^{13} T^{57} +$$$$15\!\cdots\!51$$$$p^{14} T^{58} +$$$$13\!\cdots\!33$$$$p^{15} T^{59} +$$$$11\!\cdots\!71$$$$p^{16} T^{60} +$$$$97\!\cdots\!20$$$$p^{17} T^{61} +$$$$85\!\cdots\!10$$$$p^{18} T^{62} +$$$$67\!\cdots\!34$$$$p^{19} T^{63} +$$$$56\!\cdots\!31$$$$p^{20} T^{64} +$$$$43\!\cdots\!99$$$$p^{21} T^{65} +$$$$35\!\cdots\!58$$$$p^{22} T^{66} +$$$$25\!\cdots\!81$$$$p^{23} T^{67} +$$$$19\!\cdots\!56$$$$p^{24} T^{68} +$$$$13\!\cdots\!32$$$$p^{25} T^{69} +$$$$10\!\cdots\!45$$$$p^{26} T^{70} +$$$$65\!\cdots\!09$$$$p^{27} T^{71} +$$$$46\!\cdots\!30$$$$p^{28} T^{72} +$$$$27\!\cdots\!47$$$$p^{29} T^{73} + 19306783774357687228 p^{30} T^{74} + 1056824922943738031 p^{31} T^{75} + 69508108424064762 p^{32} T^{76} + 3433863156080202 p^{33} T^{77} + 214879531214828 p^{34} T^{78} + 9345105080115 p^{35} T^{79} + 555021631313 p^{36} T^{80} + 20464453312 p^{37} T^{81} + 1151053135 p^{38} T^{82} + 33830358 p^{39} T^{83} + 1799315 p^{40} T^{84} + 37551 p^{41} T^{85} + 1888 p^{42} T^{86} + 21 p^{43} T^{87} + p^{44} T^{88}$$
97 $$1 + 67 T + 4372 T^{2} + 192321 T^{3} + 7857390 T^{4} + 268127151 T^{5} + 8538090768 T^{6} + 243676694469 T^{7} + 6552315612526 T^{8} + 163120727981891 T^{9} + 3859060136978518 T^{10} + 86056588213314313 T^{11} + 1836351955027747322 T^{12} + 37351612763405598293 T^{13} +$$$$73\!\cdots\!18$$$$T^{14} +$$$$13\!\cdots\!27$$$$T^{15} +$$$$24\!\cdots\!13$$$$T^{16} +$$$$43\!\cdots\!12$$$$T^{17} +$$$$74\!\cdots\!43$$$$T^{18} +$$$$12\!\cdots\!50$$$$T^{19} +$$$$19\!\cdots\!69$$$$T^{20} +$$$$30\!\cdots\!92$$$$T^{21} +$$$$48\!\cdots\!18$$$$p T^{22} +$$$$69\!\cdots\!75$$$$T^{23} +$$$$10\!\cdots\!38$$$$T^{24} +$$$$14\!\cdots\!82$$$$T^{25} +$$$$19\!\cdots\!50$$$$T^{26} +$$$$26\!\cdots\!37$$$$T^{27} +$$$$35\!\cdots\!51$$$$T^{28} +$$$$45\!\cdots\!71$$$$T^{29} +$$$$57\!\cdots\!81$$$$T^{30} +$$$$72\!\cdots\!53$$$$T^{31} +$$$$88\!\cdots\!41$$$$T^{32} +$$$$10\!\cdots\!98$$$$T^{33} +$$$$12\!\cdots\!68$$$$T^{34} +$$$$14\!\cdots\!02$$$$T^{35} +$$$$16\!\cdots\!99$$$$T^{36} +$$$$18\!\cdots\!96$$$$T^{37} +$$$$20\!\cdots\!45$$$$T^{38} +$$$$21\!\cdots\!27$$$$T^{39} +$$$$23\!\cdots\!08$$$$T^{40} +$$$$24\!\cdots\!62$$$$T^{41} +$$$$25\!\cdots\!74$$$$T^{42} +$$$$25\!\cdots\!41$$$$T^{43} +$$$$25\!\cdots\!14$$$$T^{44} +$$$$25\!\cdots\!41$$$$p T^{45} +$$$$25\!\cdots\!74$$$$p^{2} T^{46} +$$$$24\!\cdots\!62$$$$p^{3} T^{47} +$$$$23\!\cdots\!08$$$$p^{4} T^{48} +$$$$21\!\cdots\!27$$$$p^{5} T^{49} +$$$$20\!\cdots\!45$$$$p^{6} T^{50} +$$$$18\!\cdots\!96$$$$p^{7} T^{51} +$$$$16\!\cdots\!99$$$$p^{8} T^{52} +$$$$14\!\cdots\!02$$$$p^{9} T^{53} +$$$$12\!\cdots\!68$$$$p^{10} T^{54} +$$$$10\!\cdots\!98$$$$p^{11} T^{55} +$$$$88\!\cdots\!41$$$$p^{12} T^{56} +$$$$72\!\cdots\!53$$$$p^{13} T^{57} +$$$$57\!\cdots\!81$$$$p^{14} T^{58} +$$$$45\!\cdots\!71$$$$p^{15} T^{59} +$$$$35\!\cdots\!51$$$$p^{16} T^{60} +$$$$26\!\cdots\!37$$$$p^{17} T^{61} +$$$$19\!\cdots\!50$$$$p^{18} T^{62} +$$$$14\!\cdots\!82$$$$p^{19} T^{63} +$$$$10\!\cdots\!38$$$$p^{20} T^{64} +$$$$69\!\cdots\!75$$$$p^{21} T^{65} +$$$$48\!\cdots\!18$$$$p^{23} T^{66} +$$$$30\!\cdots\!92$$$$p^{23} T^{67} +$$$$19\!\cdots\!69$$$$p^{24} T^{68} +$$$$12\!\cdots\!50$$$$p^{25} T^{69} +$$$$74\!\cdots\!43$$$$p^{26} T^{70} +$$$$43\!\cdots\!12$$$$p^{27} T^{71} +$$$$24\!\cdots\!13$$$$p^{28} T^{72} +$$$$13\!\cdots\!27$$$$p^{29} T^{73} +$$$$73\!\cdots\!18$$$$p^{30} T^{74} + 37351612763405598293 p^{31} T^{75} + 1836351955027747322 p^{32} T^{76} + 86056588213314313 p^{33} T^{77} + 3859060136978518 p^{34} T^{78} + 163120727981891 p^{35} T^{79} + 6552315612526 p^{36} T^{80} + 243676694469 p^{37} T^{81} + 8538090768 p^{38} T^{82} + 268127151 p^{39} T^{83} + 7857390 p^{40} T^{84} + 192321 p^{41} T^{85} + 4372 p^{42} T^{86} + 67 p^{43} T^{87} + p^{44} T^{88}$$
\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{88} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}