# Properties

 Degree $56$ Conductor $1.004\times 10^{59}$ Sign $1$ Motivic weight $2$ Primitive no Self-dual yes Analytic rank $0$

# Origins of factors

## Dirichlet series

 L(s)  = 1 + 4·3-s − 4·5-s + 4·7-s + 6·9-s + 4·11-s − 4·13-s − 16·15-s + 4·19-s + 16·21-s + 68·23-s + 6·25-s + 36·27-s − 4·29-s + 16·33-s − 16·35-s − 4·37-s − 16·39-s − 4·41-s − 92·43-s − 24·45-s + 8·47-s + 8·49-s − 164·53-s − 16·55-s + 16·57-s − 124·59-s − 68·61-s + ⋯
 L(s)  = 1 + 4/3·3-s − 4/5·5-s + 4/7·7-s + 2/3·9-s + 4/11·11-s − 0.307·13-s − 1.06·15-s + 4/19·19-s + 0.761·21-s + 2.95·23-s + 6/25·25-s + 4/3·27-s − 0.137·29-s + 0.484·33-s − 0.457·35-s − 0.108·37-s − 0.410·39-s − 0.0975·41-s − 2.13·43-s − 0.533·45-s + 8/47·47-s + 8/49·49-s − 3.09·53-s − 0.290·55-s + 0.280·57-s − 2.10·59-s − 1.11·61-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{196}\right)^{s/2} \, \Gamma_{\C}(s)^{28} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{196}\right)^{s/2} \, \Gamma_{\C}(s+1)^{28} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}

## Invariants

 Degree: $$56$$ Conductor: $$2^{196}$$ Sign: $1$ Motivic weight: $$2$$ Character: induced by $\chi_{128} (1, \cdot )$ Primitive: no Self-dual: yes Analytic rank: $$0$$ Selberg data: $$(56,\ 2^{196} ,\ ( \ : [1]^{28} ),\ 1 )$$

## Particular Values

 $$L(\frac{3}{2})$$ $$\approx$$ $$1.20538$$ $$L(\frac12)$$ $$\approx$$ $$1.20538$$ $$L(2)$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad2 $$1$$
good3 $$1 - 4 T + 10 T^{2} - 52 T^{3} + 2 p^{4} T^{4} - 52 T^{5} - 662 T^{6} + 1364 p T^{7} - 26245 T^{8} + 104504 T^{9} - 26332 p^{2} T^{10} + 389912 T^{11} - 302092 T^{12} - 1820216 p T^{13} + 11800108 p T^{14} - 116049416 T^{15} + 341895529 T^{16} - 834464092 T^{17} + 845214806 T^{18} + 4596163348 T^{19} - 9906234710 p T^{20} + 38972759708 p T^{21} - 390403683722 T^{22} + 1007116990948 T^{23} - 1306714409509 T^{24} - 1366587470960 T^{25} + 17418041503544 T^{26} - 92007761930032 T^{27} + 332896726659032 T^{28} - 92007761930032 p^{2} T^{29} + 17418041503544 p^{4} T^{30} - 1366587470960 p^{6} T^{31} - 1306714409509 p^{8} T^{32} + 1007116990948 p^{10} T^{33} - 390403683722 p^{12} T^{34} + 38972759708 p^{15} T^{35} - 9906234710 p^{17} T^{36} + 4596163348 p^{18} T^{37} + 845214806 p^{20} T^{38} - 834464092 p^{22} T^{39} + 341895529 p^{24} T^{40} - 116049416 p^{26} T^{41} + 11800108 p^{29} T^{42} - 1820216 p^{31} T^{43} - 302092 p^{32} T^{44} + 389912 p^{34} T^{45} - 26332 p^{38} T^{46} + 104504 p^{38} T^{47} - 26245 p^{40} T^{48} + 1364 p^{43} T^{49} - 662 p^{44} T^{50} - 52 p^{46} T^{51} + 2 p^{52} T^{52} - 52 p^{50} T^{53} + 10 p^{52} T^{54} - 4 p^{54} T^{55} + p^{56} T^{56}$$
5 $$1 + 4 T + 2 p T^{2} - 12 T^{3} - 94 T^{4} + 4 p^{3} T^{5} + 938 T^{6} + 46724 T^{7} - 266469 T^{8} - 291608 p T^{9} - 5625596 T^{10} - 40645592 T^{11} - 91628364 T^{12} - 1781075096 T^{13} - 2864474684 T^{14} - 34472996024 T^{15} + 3350654897 p^{2} T^{16} + 387818789404 T^{17} + 443469243222 T^{18} + 10790165594348 T^{19} - 35916754299138 T^{20} + 462914656796332 T^{21} - 431429275333322 T^{22} + 11185946366012124 T^{23} + 32508880692810971 T^{24} + 21996555420373488 T^{25} + 794534008776349624 T^{26} - 263431193473525968 p^{2} T^{27} + 25161869921157155928 T^{28} - 263431193473525968 p^{4} T^{29} + 794534008776349624 p^{4} T^{30} + 21996555420373488 p^{6} T^{31} + 32508880692810971 p^{8} T^{32} + 11185946366012124 p^{10} T^{33} - 431429275333322 p^{12} T^{34} + 462914656796332 p^{14} T^{35} - 35916754299138 p^{16} T^{36} + 10790165594348 p^{18} T^{37} + 443469243222 p^{20} T^{38} + 387818789404 p^{22} T^{39} + 3350654897 p^{26} T^{40} - 34472996024 p^{26} T^{41} - 2864474684 p^{28} T^{42} - 1781075096 p^{30} T^{43} - 91628364 p^{32} T^{44} - 40645592 p^{34} T^{45} - 5625596 p^{36} T^{46} - 291608 p^{39} T^{47} - 266469 p^{40} T^{48} + 46724 p^{42} T^{49} + 938 p^{44} T^{50} + 4 p^{49} T^{51} - 94 p^{48} T^{52} - 12 p^{50} T^{53} + 2 p^{53} T^{54} + 4 p^{54} T^{55} + p^{56} T^{56}$$
7 $$1 - 4 T + 8 T^{2} - 844 T^{3} + 142 p T^{4} - 3036 T^{5} + 51480 p T^{6} - 609844 T^{7} + 9676763 T^{8} - 94895352 T^{9} + 463286448 T^{10} - 4759916008 T^{11} + 15226347636 T^{12} - 225537389608 T^{13} + 189162854992 p T^{14} - 9435650886840 T^{15} - 3662352244311 T^{16} + 47964800936212 T^{17} + 657850810512296 p T^{18} + 50618705885725692 T^{19} - 5637566014622638 p T^{20} - 1249426904106242132 T^{21} - 22194382343104241224 T^{22} + 27336166747157725348 T^{23} - 7951991184565238549 p^{2} T^{24} +$$$$80\!\cdots\!24$$$$p T^{25} -$$$$30\!\cdots\!20$$$$T^{26} +$$$$28\!\cdots\!64$$$$T^{27} -$$$$10\!\cdots\!32$$$$T^{28} +$$$$28\!\cdots\!64$$$$p^{2} T^{29} -$$$$30\!\cdots\!20$$$$p^{4} T^{30} +$$$$80\!\cdots\!24$$$$p^{7} T^{31} - 7951991184565238549 p^{10} T^{32} + 27336166747157725348 p^{10} T^{33} - 22194382343104241224 p^{12} T^{34} - 1249426904106242132 p^{14} T^{35} - 5637566014622638 p^{17} T^{36} + 50618705885725692 p^{18} T^{37} + 657850810512296 p^{21} T^{38} + 47964800936212 p^{22} T^{39} - 3662352244311 p^{24} T^{40} - 9435650886840 p^{26} T^{41} + 189162854992 p^{29} T^{42} - 225537389608 p^{30} T^{43} + 15226347636 p^{32} T^{44} - 4759916008 p^{34} T^{45} + 463286448 p^{36} T^{46} - 94895352 p^{38} T^{47} + 9676763 p^{40} T^{48} - 609844 p^{42} T^{49} + 51480 p^{45} T^{50} - 3036 p^{46} T^{51} + 142 p^{49} T^{52} - 844 p^{50} T^{53} + 8 p^{52} T^{54} - 4 p^{54} T^{55} + p^{56} T^{56}$$
11 $$1 - 4 T + 106 T^{2} + 524 T^{3} + 1890 T^{4} - 56500 T^{5} - 2477430 T^{6} + 2831804 T^{7} - 485405765 T^{8} - 121330008 p T^{9} - 16124678396 T^{10} - 351621465192 T^{11} + 4810836128884 T^{12} - 1122713383928 p T^{13} + 41655264441036 p T^{14} + 6365499089805432 T^{15} + 28186548831723497 T^{16} + 902427850069482916 T^{17} - 1679784019800395530 T^{18} + 88242810381789933268 T^{19} - 19072453056229186694 p T^{20} -$$$$77\!\cdots\!72$$$$T^{21} +$$$$18\!\cdots\!86$$$$T^{22} -$$$$22\!\cdots\!04$$$$T^{23} +$$$$55\!\cdots\!39$$$$T^{24} +$$$$43\!\cdots\!12$$$$T^{25} -$$$$20\!\cdots\!80$$$$T^{26} +$$$$11\!\cdots\!52$$$$T^{27} -$$$$39\!\cdots\!56$$$$T^{28} +$$$$11\!\cdots\!52$$$$p^{2} T^{29} -$$$$20\!\cdots\!80$$$$p^{4} T^{30} +$$$$43\!\cdots\!12$$$$p^{6} T^{31} +$$$$55\!\cdots\!39$$$$p^{8} T^{32} -$$$$22\!\cdots\!04$$$$p^{10} T^{33} +$$$$18\!\cdots\!86$$$$p^{12} T^{34} -$$$$77\!\cdots\!72$$$$p^{14} T^{35} - 19072453056229186694 p^{17} T^{36} + 88242810381789933268 p^{18} T^{37} - 1679784019800395530 p^{20} T^{38} + 902427850069482916 p^{22} T^{39} + 28186548831723497 p^{24} T^{40} + 6365499089805432 p^{26} T^{41} + 41655264441036 p^{29} T^{42} - 1122713383928 p^{31} T^{43} + 4810836128884 p^{32} T^{44} - 351621465192 p^{34} T^{45} - 16124678396 p^{36} T^{46} - 121330008 p^{39} T^{47} - 485405765 p^{40} T^{48} + 2831804 p^{42} T^{49} - 2477430 p^{44} T^{50} - 56500 p^{46} T^{51} + 1890 p^{48} T^{52} + 524 p^{50} T^{53} + 106 p^{52} T^{54} - 4 p^{54} T^{55} + p^{56} T^{56}$$
13 $$1 + 4 T + 10 T^{2} + 2740 T^{3} + 10914 T^{4} - 202508 T^{5} - 1998102 T^{6} + 40271940 T^{7} - 438003749 T^{8} - 8261748344 T^{9} + 40947033604 T^{10} + 822539868840 T^{11} - 2104580462668 T^{12} - 154177655740312 T^{13} + 4150835490424644 T^{14} + 35615346534284744 T^{15} + 221854271492021449 T^{16} - 413649847941221092 T^{17} +$$$$13\!\cdots\!94$$$$T^{18} +$$$$20\!\cdots\!28$$$$T^{19} -$$$$16\!\cdots\!74$$$$T^{20} +$$$$12\!\cdots\!12$$$$T^{21} +$$$$20\!\cdots\!02$$$$T^{22} -$$$$10\!\cdots\!44$$$$T^{23} -$$$$47\!\cdots\!57$$$$T^{24} +$$$$41\!\cdots\!04$$$$T^{25} +$$$$55\!\cdots\!76$$$$T^{26} -$$$$71\!\cdots\!76$$$$T^{27} +$$$$11\!\cdots\!24$$$$T^{28} -$$$$71\!\cdots\!76$$$$p^{2} T^{29} +$$$$55\!\cdots\!76$$$$p^{4} T^{30} +$$$$41\!\cdots\!04$$$$p^{6} T^{31} -$$$$47\!\cdots\!57$$$$p^{8} T^{32} -$$$$10\!\cdots\!44$$$$p^{10} T^{33} +$$$$20\!\cdots\!02$$$$p^{12} T^{34} +$$$$12\!\cdots\!12$$$$p^{14} T^{35} -$$$$16\!\cdots\!74$$$$p^{16} T^{36} +$$$$20\!\cdots\!28$$$$p^{18} T^{37} +$$$$13\!\cdots\!94$$$$p^{20} T^{38} - 413649847941221092 p^{22} T^{39} + 221854271492021449 p^{24} T^{40} + 35615346534284744 p^{26} T^{41} + 4150835490424644 p^{28} T^{42} - 154177655740312 p^{30} T^{43} - 2104580462668 p^{32} T^{44} + 822539868840 p^{34} T^{45} + 40947033604 p^{36} T^{46} - 8261748344 p^{38} T^{47} - 438003749 p^{40} T^{48} + 40271940 p^{42} T^{49} - 1998102 p^{44} T^{50} - 202508 p^{46} T^{51} + 10914 p^{48} T^{52} + 2740 p^{50} T^{53} + 10 p^{52} T^{54} + 4 p^{54} T^{55} + p^{56} T^{56}$$
17 $$1 - 4348 T^{2} + 9363834 T^{4} - 13373503020 T^{6} + 14302771061115 T^{8} - 12253150019337000 T^{10} + 8774712741770357156 T^{12} -$$$$54\!\cdots\!84$$$$T^{14} +$$$$29\!\cdots\!37$$$$T^{16} -$$$$14\!\cdots\!40$$$$T^{18} +$$$$61\!\cdots\!90$$$$T^{20} -$$$$24\!\cdots\!44$$$$T^{22} +$$$$86\!\cdots\!75$$$$T^{24} -$$$$28\!\cdots\!16$$$$T^{26} +$$$$85\!\cdots\!96$$$$T^{28} -$$$$28\!\cdots\!16$$$$p^{4} T^{30} +$$$$86\!\cdots\!75$$$$p^{8} T^{32} -$$$$24\!\cdots\!44$$$$p^{12} T^{34} +$$$$61\!\cdots\!90$$$$p^{16} T^{36} -$$$$14\!\cdots\!40$$$$p^{20} T^{38} +$$$$29\!\cdots\!37$$$$p^{24} T^{40} -$$$$54\!\cdots\!84$$$$p^{28} T^{42} + 8774712741770357156 p^{32} T^{44} - 12253150019337000 p^{36} T^{46} + 14302771061115 p^{40} T^{48} - 13373503020 p^{44} T^{50} + 9363834 p^{48} T^{52} - 4348 p^{52} T^{54} + p^{56} T^{56}$$
19 $$1 - 4 T + 714 T^{2} - 7412 T^{3} + 273186 T^{4} - 3452916 T^{5} - 20323542 T^{6} - 965215492 T^{7} - 89037798661 T^{8} + 785372418744 T^{9} - 51192199977916 T^{10} + 850130807216920 T^{11} - 15994445965485836 T^{12} + 431294796813599832 T^{13} + 866625201212778884 T^{14} + 91329818783853744504 T^{15} +$$$$27\!\cdots\!37$$$$T^{16} -$$$$20\!\cdots\!52$$$$p T^{17} +$$$$15\!\cdots\!66$$$$T^{18} -$$$$40\!\cdots\!84$$$$T^{19} +$$$$45\!\cdots\!34$$$$T^{20} -$$$$18\!\cdots\!68$$$$T^{21} +$$$$47\!\cdots\!50$$$$T^{22} -$$$$29\!\cdots\!08$$$$T^{23} -$$$$25\!\cdots\!57$$$$T^{24} +$$$$11\!\cdots\!76$$$$T^{25} -$$$$21\!\cdots\!36$$$$T^{26} +$$$$10\!\cdots\!20$$$$T^{27} -$$$$86\!\cdots\!48$$$$T^{28} +$$$$10\!\cdots\!20$$$$p^{2} T^{29} -$$$$21\!\cdots\!36$$$$p^{4} T^{30} +$$$$11\!\cdots\!76$$$$p^{6} T^{31} -$$$$25\!\cdots\!57$$$$p^{8} T^{32} -$$$$29\!\cdots\!08$$$$p^{10} T^{33} +$$$$47\!\cdots\!50$$$$p^{12} T^{34} -$$$$18\!\cdots\!68$$$$p^{14} T^{35} +$$$$45\!\cdots\!34$$$$p^{16} T^{36} -$$$$40\!\cdots\!84$$$$p^{18} T^{37} +$$$$15\!\cdots\!66$$$$p^{20} T^{38} -$$$$20\!\cdots\!52$$$$p^{23} T^{39} +$$$$27\!\cdots\!37$$$$p^{24} T^{40} + 91329818783853744504 p^{26} T^{41} + 866625201212778884 p^{28} T^{42} + 431294796813599832 p^{30} T^{43} - 15994445965485836 p^{32} T^{44} + 850130807216920 p^{34} T^{45} - 51192199977916 p^{36} T^{46} + 785372418744 p^{38} T^{47} - 89037798661 p^{40} T^{48} - 965215492 p^{42} T^{49} - 20323542 p^{44} T^{50} - 3452916 p^{46} T^{51} + 273186 p^{48} T^{52} - 7412 p^{50} T^{53} + 714 p^{52} T^{54} - 4 p^{54} T^{55} + p^{56} T^{56}$$
23 $$1 - 68 T + 2312 T^{2} - 67468 T^{3} + 1818594 T^{4} - 33584540 T^{5} + 355124904 T^{6} + 6433822732 T^{7} - 673978345381 T^{8} + 23516816703880 T^{9} - 556850109584720 T^{10} + 12237631186275480 T^{11} - 160281848565966220 T^{12} - 1764229102924708008 T^{13} +$$$$14\!\cdots\!60$$$$T^{14} -$$$$57\!\cdots\!88$$$$T^{15} +$$$$16\!\cdots\!01$$$$T^{16} -$$$$29\!\cdots\!80$$$$T^{17} +$$$$35\!\cdots\!12$$$$T^{18} +$$$$10\!\cdots\!44$$$$T^{19} -$$$$34\!\cdots\!98$$$$T^{20} +$$$$56\!\cdots\!96$$$$p T^{21} -$$$$31\!\cdots\!88$$$$T^{22} +$$$$67\!\cdots\!84$$$$T^{23} -$$$$90\!\cdots\!45$$$$T^{24} -$$$$30\!\cdots\!88$$$$T^{25} +$$$$47\!\cdots\!48$$$$T^{26} -$$$$19\!\cdots\!00$$$$T^{27} +$$$$56\!\cdots\!56$$$$T^{28} -$$$$19\!\cdots\!00$$$$p^{2} T^{29} +$$$$47\!\cdots\!48$$$$p^{4} T^{30} -$$$$30\!\cdots\!88$$$$p^{6} T^{31} -$$$$90\!\cdots\!45$$$$p^{8} T^{32} +$$$$67\!\cdots\!84$$$$p^{10} T^{33} -$$$$31\!\cdots\!88$$$$p^{12} T^{34} +$$$$56\!\cdots\!96$$$$p^{15} T^{35} -$$$$34\!\cdots\!98$$$$p^{16} T^{36} +$$$$10\!\cdots\!44$$$$p^{18} T^{37} +$$$$35\!\cdots\!12$$$$p^{20} T^{38} -$$$$29\!\cdots\!80$$$$p^{22} T^{39} +$$$$16\!\cdots\!01$$$$p^{24} T^{40} -$$$$57\!\cdots\!88$$$$p^{26} T^{41} +$$$$14\!\cdots\!60$$$$p^{28} T^{42} - 1764229102924708008 p^{30} T^{43} - 160281848565966220 p^{32} T^{44} + 12237631186275480 p^{34} T^{45} - 556850109584720 p^{36} T^{46} + 23516816703880 p^{38} T^{47} - 673978345381 p^{40} T^{48} + 6433822732 p^{42} T^{49} + 355124904 p^{44} T^{50} - 33584540 p^{46} T^{51} + 1818594 p^{48} T^{52} - 67468 p^{50} T^{53} + 2312 p^{52} T^{54} - 68 p^{54} T^{55} + p^{56} T^{56}$$
29 $$1 + 4 T + 874 T^{2} + 17204 T^{3} + 436834 T^{4} + 36632564 T^{5} + 384900810 T^{6} + 43063510980 T^{7} + 1703369788251 T^{8} + 26598683470216 T^{9} + 1992147286114756 T^{10} + 43072602208091560 T^{11} + 1562672686617477940 T^{12} + 67740241856123696232 T^{13} +$$$$13\!\cdots\!00$$$$T^{14} +$$$$72\!\cdots\!40$$$$T^{15} +$$$$17\!\cdots\!77$$$$T^{16} +$$$$51\!\cdots\!88$$$$T^{17} +$$$$21\!\cdots\!34$$$$T^{18} +$$$$51\!\cdots\!88$$$$T^{19} +$$$$18\!\cdots\!46$$$$T^{20} +$$$$57\!\cdots\!76$$$$T^{21} +$$$$16\!\cdots\!62$$$$T^{22} +$$$$56\!\cdots\!76$$$$T^{23} +$$$$14\!\cdots\!99$$$$T^{24} +$$$$45\!\cdots\!40$$$$T^{25} +$$$$14\!\cdots\!16$$$$T^{26} +$$$$36\!\cdots\!12$$$$T^{27} +$$$$12\!\cdots\!32$$$$T^{28} +$$$$36\!\cdots\!12$$$$p^{2} T^{29} +$$$$14\!\cdots\!16$$$$p^{4} T^{30} +$$$$45\!\cdots\!40$$$$p^{6} T^{31} +$$$$14\!\cdots\!99$$$$p^{8} T^{32} +$$$$56\!\cdots\!76$$$$p^{10} T^{33} +$$$$16\!\cdots\!62$$$$p^{12} T^{34} +$$$$57\!\cdots\!76$$$$p^{14} T^{35} +$$$$18\!\cdots\!46$$$$p^{16} T^{36} +$$$$51\!\cdots\!88$$$$p^{18} T^{37} +$$$$21\!\cdots\!34$$$$p^{20} T^{38} +$$$$51\!\cdots\!88$$$$p^{22} T^{39} +$$$$17\!\cdots\!77$$$$p^{24} T^{40} +$$$$72\!\cdots\!40$$$$p^{26} T^{41} +$$$$13\!\cdots\!00$$$$p^{28} T^{42} + 67740241856123696232 p^{30} T^{43} + 1562672686617477940 p^{32} T^{44} + 43072602208091560 p^{34} T^{45} + 1992147286114756 p^{36} T^{46} + 26598683470216 p^{38} T^{47} + 1703369788251 p^{40} T^{48} + 43063510980 p^{42} T^{49} + 384900810 p^{44} T^{50} + 36632564 p^{46} T^{51} + 436834 p^{48} T^{52} + 17204 p^{50} T^{53} + 874 p^{52} T^{54} + 4 p^{54} T^{55} + p^{56} T^{56}$$
31 $$1 - 14876 T^{2} + 109353850 T^{4} - 531000467660 T^{6} + 1921146182996219 T^{8} - 5537763963798420456 T^{10} +$$$$13\!\cdots\!12$$$$T^{12} -$$$$27\!\cdots\!68$$$$T^{14} +$$$$49\!\cdots\!33$$$$T^{16} -$$$$78\!\cdots\!56$$$$T^{18} +$$$$11\!\cdots\!54$$$$T^{20} -$$$$14\!\cdots\!28$$$$T^{22} +$$$$17\!\cdots\!55$$$$T^{24} -$$$$19\!\cdots\!84$$$$T^{26} +$$$$19\!\cdots\!32$$$$p^{2} T^{28} -$$$$19\!\cdots\!84$$$$p^{4} T^{30} +$$$$17\!\cdots\!55$$$$p^{8} T^{32} -$$$$14\!\cdots\!28$$$$p^{12} T^{34} +$$$$11\!\cdots\!54$$$$p^{16} T^{36} -$$$$78\!\cdots\!56$$$$p^{20} T^{38} +$$$$49\!\cdots\!33$$$$p^{24} T^{40} -$$$$27\!\cdots\!68$$$$p^{28} T^{42} +$$$$13\!\cdots\!12$$$$p^{32} T^{44} - 5537763963798420456 p^{36} T^{46} + 1921146182996219 p^{40} T^{48} - 531000467660 p^{44} T^{50} + 109353850 p^{48} T^{52} - 14876 p^{52} T^{54} + p^{56} T^{56}$$
37 $$1 + 4 T + 3050 T^{2} - 165004 T^{3} + 106554 p T^{4} - 531533708 T^{5} + 21845067402 T^{6} - 699331240572 T^{7} + 56559452783131 T^{8} - 1994623508619128 T^{9} + 62708147115168196 T^{10} - 4299941341962325720 T^{11} +$$$$12\!\cdots\!08$$$$T^{12} -$$$$38\!\cdots\!32$$$$T^{13} +$$$$23\!\cdots\!00$$$$T^{14} -$$$$49\!\cdots\!00$$$$T^{15} +$$$$18\!\cdots\!93$$$$T^{16} -$$$$10\!\cdots\!12$$$$T^{17} +$$$$16\!\cdots\!38$$$$T^{18} -$$$$89\!\cdots\!68$$$$T^{19} +$$$$46\!\cdots\!46$$$$T^{20} -$$$$75\!\cdots\!20$$$$T^{21} +$$$$64\!\cdots\!14$$$$T^{22} -$$$$25\!\cdots\!80$$$$T^{23} +$$$$67\!\cdots\!15$$$$T^{24} -$$$$48\!\cdots\!36$$$$T^{25} +$$$$15\!\cdots\!60$$$$T^{26} -$$$$51\!\cdots\!88$$$$T^{27} +$$$$28\!\cdots\!04$$$$T^{28} -$$$$51\!\cdots\!88$$$$p^{2} T^{29} +$$$$15\!\cdots\!60$$$$p^{4} T^{30} -$$$$48\!\cdots\!36$$$$p^{6} T^{31} +$$$$67\!\cdots\!15$$$$p^{8} T^{32} -$$$$25\!\cdots\!80$$$$p^{10} T^{33} +$$$$64\!\cdots\!14$$$$p^{12} T^{34} -$$$$75\!\cdots\!20$$$$p^{14} T^{35} +$$$$46\!\cdots\!46$$$$p^{16} T^{36} -$$$$89\!\cdots\!68$$$$p^{18} T^{37} +$$$$16\!\cdots\!38$$$$p^{20} T^{38} -$$$$10\!\cdots\!12$$$$p^{22} T^{39} +$$$$18\!\cdots\!93$$$$p^{24} T^{40} -$$$$49\!\cdots\!00$$$$p^{26} T^{41} +$$$$23\!\cdots\!00$$$$p^{28} T^{42} -$$$$38\!\cdots\!32$$$$p^{30} T^{43} +$$$$12\!\cdots\!08$$$$p^{32} T^{44} - 4299941341962325720 p^{34} T^{45} + 62708147115168196 p^{36} T^{46} - 1994623508619128 p^{38} T^{47} + 56559452783131 p^{40} T^{48} - 699331240572 p^{42} T^{49} + 21845067402 p^{44} T^{50} - 531533708 p^{46} T^{51} + 106554 p^{49} T^{52} - 165004 p^{50} T^{53} + 3050 p^{52} T^{54} + 4 p^{54} T^{55} + p^{56} T^{56}$$
41 $$1 + 4 T + 8 T^{2} - 146740 T^{3} + 5299010 T^{4} + 39014236 T^{5} + 10879978664 T^{6} - 615549292748 T^{7} + 16970020586075 T^{8} - 827024534256008 T^{9} + 29989784481138096 T^{10} - 1999131777934499992 T^{11} + 76703261367724022836 T^{12} -$$$$33\!\cdots\!40$$$$T^{13} +$$$$93\!\cdots\!36$$$$T^{14} -$$$$41\!\cdots\!08$$$$T^{15} +$$$$19\!\cdots\!29$$$$T^{16} -$$$$15\!\cdots\!68$$$$T^{17} +$$$$15\!\cdots\!60$$$$T^{18} -$$$$13\!\cdots\!68$$$$T^{19} +$$$$95\!\cdots\!94$$$$T^{20} -$$$$27\!\cdots\!16$$$$T^{21} +$$$$16\!\cdots\!80$$$$T^{22} -$$$$64\!\cdots\!08$$$$T^{23} +$$$$14\!\cdots\!99$$$$T^{24} -$$$$11\!\cdots\!08$$$$T^{25} +$$$$77\!\cdots\!68$$$$T^{26} -$$$$39\!\cdots\!24$$$$p T^{27} +$$$$42\!\cdots\!08$$$$p^{2} T^{28} -$$$$39\!\cdots\!24$$$$p^{3} T^{29} +$$$$77\!\cdots\!68$$$$p^{4} T^{30} -$$$$11\!\cdots\!08$$$$p^{6} T^{31} +$$$$14\!\cdots\!99$$$$p^{8} T^{32} -$$$$64\!\cdots\!08$$$$p^{10} T^{33} +$$$$16\!\cdots\!80$$$$p^{12} T^{34} -$$$$27\!\cdots\!16$$$$p^{14} T^{35} +$$$$95\!\cdots\!94$$$$p^{16} T^{36} -$$$$13\!\cdots\!68$$$$p^{18} T^{37} +$$$$15\!\cdots\!60$$$$p^{20} T^{38} -$$$$15\!\cdots\!68$$$$p^{22} T^{39} +$$$$19\!\cdots\!29$$$$p^{24} T^{40} -$$$$41\!\cdots\!08$$$$p^{26} T^{41} +$$$$93\!\cdots\!36$$$$p^{28} T^{42} -$$$$33\!\cdots\!40$$$$p^{30} T^{43} + 76703261367724022836 p^{32} T^{44} - 1999131777934499992 p^{34} T^{45} + 29989784481138096 p^{36} T^{46} - 827024534256008 p^{38} T^{47} + 16970020586075 p^{40} T^{48} - 615549292748 p^{42} T^{49} + 10879978664 p^{44} T^{50} + 39014236 p^{46} T^{51} + 5299010 p^{48} T^{52} - 146740 p^{50} T^{53} + 8 p^{52} T^{54} + 4 p^{54} T^{55} + p^{56} T^{56}$$
43 $$1 + 92 T + 13354 T^{2} + 631532 T^{3} + 52147170 T^{4} + 399598380 T^{5} + 34718379466 T^{6} - 8398618107364 T^{7} - 227729338315845 T^{8} - 31550446902888200 T^{9} - 114294784020238332 T^{10} - 33962858106170505896 T^{11} +$$$$29\!\cdots\!92$$$$T^{12} +$$$$31\!\cdots\!52$$$$T^{13} +$$$$95\!\cdots\!40$$$$T^{14} -$$$$38\!\cdots\!40$$$$T^{15} +$$$$10\!\cdots\!73$$$$T^{16} -$$$$80\!\cdots\!96$$$$T^{17} -$$$$42\!\cdots\!98$$$$p T^{18} -$$$$21\!\cdots\!32$$$$T^{19} +$$$$25\!\cdots\!50$$$$T^{20} -$$$$19\!\cdots\!96$$$$T^{21} +$$$$18\!\cdots\!18$$$$T^{22} +$$$$14\!\cdots\!40$$$$T^{23} +$$$$40\!\cdots\!23$$$$T^{24} -$$$$60\!\cdots\!48$$$$T^{25} +$$$$29\!\cdots\!20$$$$T^{26} -$$$$33\!\cdots\!60$$$$T^{27} -$$$$38\!\cdots\!80$$$$T^{28} -$$$$33\!\cdots\!60$$$$p^{2} T^{29} +$$$$29\!\cdots\!20$$$$p^{4} T^{30} -$$$$60\!\cdots\!48$$$$p^{6} T^{31} +$$$$40\!\cdots\!23$$$$p^{8} T^{32} +$$$$14\!\cdots\!40$$$$p^{10} T^{33} +$$$$18\!\cdots\!18$$$$p^{12} T^{34} -$$$$19\!\cdots\!96$$$$p^{14} T^{35} +$$$$25\!\cdots\!50$$$$p^{16} T^{36} -$$$$21\!\cdots\!32$$$$p^{18} T^{37} -$$$$42\!\cdots\!98$$$$p^{21} T^{38} -$$$$80\!\cdots\!96$$$$p^{22} T^{39} +$$$$10\!\cdots\!73$$$$p^{24} T^{40} -$$$$38\!\cdots\!40$$$$p^{26} T^{41} +$$$$95\!\cdots\!40$$$$p^{28} T^{42} +$$$$31\!\cdots\!52$$$$p^{30} T^{43} +$$$$29\!\cdots\!92$$$$p^{32} T^{44} - 33962858106170505896 p^{34} T^{45} - 114294784020238332 p^{36} T^{46} - 31550446902888200 p^{38} T^{47} - 227729338315845 p^{40} T^{48} - 8398618107364 p^{42} T^{49} + 34718379466 p^{44} T^{50} + 399598380 p^{46} T^{51} + 52147170 p^{48} T^{52} + 631532 p^{50} T^{53} + 13354 p^{52} T^{54} + 92 p^{54} T^{55} + p^{56} T^{56}$$
47 $$( 1 - 4 T + 17306 T^{2} - 201684 T^{3} + 150597883 T^{4} - 2640471640 T^{5} + 883349166436 T^{6} - 19014036802584 T^{7} + 3915610933481929 T^{8} - 92637563420056524 T^{9} + 13859866263483156230 T^{10} -$$$$33\!\cdots\!92$$$$T^{11} +$$$$40\!\cdots\!51$$$$T^{12} -$$$$93\!\cdots\!08$$$$T^{13} +$$$$97\!\cdots\!92$$$$T^{14} -$$$$93\!\cdots\!08$$$$p^{2} T^{15} +$$$$40\!\cdots\!51$$$$p^{4} T^{16} -$$$$33\!\cdots\!92$$$$p^{6} T^{17} + 13859866263483156230 p^{8} T^{18} - 92637563420056524 p^{10} T^{19} + 3915610933481929 p^{12} T^{20} - 19014036802584 p^{14} T^{21} + 883349166436 p^{16} T^{22} - 2640471640 p^{18} T^{23} + 150597883 p^{20} T^{24} - 201684 p^{22} T^{25} + 17306 p^{24} T^{26} - 4 p^{26} T^{27} + p^{28} T^{28} )^{2}$$
53 $$1 + 164 T + 12778 T^{2} + 477588 T^{3} - 2865310 T^{4} - 1487543276 T^{5} - 40799035126 T^{6} + 5094170773668 T^{7} + 519330767244699 T^{8} + 17287941298996936 T^{9} - 329950203771387964 T^{10} - 49953512556108287384 T^{11} +$$$$15\!\cdots\!84$$$$T^{12} +$$$$28\!\cdots\!48$$$$T^{13} +$$$$18\!\cdots\!04$$$$T^{14} +$$$$31\!\cdots\!60$$$$T^{15} -$$$$30\!\cdots\!35$$$$T^{16} -$$$$21\!\cdots\!08$$$$T^{17} +$$$$78\!\cdots\!66$$$$T^{18} +$$$$81\!\cdots\!36$$$$T^{19} +$$$$42\!\cdots\!14$$$$T^{20} +$$$$81\!\cdots\!04$$$$T^{21} -$$$$99\!\cdots\!30$$$$T^{22} -$$$$43\!\cdots\!00$$$$T^{23} +$$$$13\!\cdots\!39$$$$T^{24} +$$$$21\!\cdots\!92$$$$T^{25} +$$$$69\!\cdots\!20$$$$T^{26} -$$$$21\!\cdots\!56$$$$T^{27} -$$$$26\!\cdots\!48$$$$T^{28} -$$$$21\!\cdots\!56$$$$p^{2} T^{29} +$$$$69\!\cdots\!20$$$$p^{4} T^{30} +$$$$21\!\cdots\!92$$$$p^{6} T^{31} +$$$$13\!\cdots\!39$$$$p^{8} T^{32} -$$$$43\!\cdots\!00$$$$p^{10} T^{33} -$$$$99\!\cdots\!30$$$$p^{12} T^{34} +$$$$81\!\cdots\!04$$$$p^{14} T^{35} +$$$$42\!\cdots\!14$$$$p^{16} T^{36} +$$$$81\!\cdots\!36$$$$p^{18} T^{37} +$$$$78\!\cdots\!66$$$$p^{20} T^{38} -$$$$21\!\cdots\!08$$$$p^{22} T^{39} -$$$$30\!\cdots\!35$$$$p^{24} T^{40} +$$$$31\!\cdots\!60$$$$p^{26} T^{41} +$$$$18\!\cdots\!04$$$$p^{28} T^{42} +$$$$28\!\cdots\!48$$$$p^{30} T^{43} +$$$$15\!\cdots\!84$$$$p^{32} T^{44} - 49953512556108287384 p^{34} T^{45} - 329950203771387964 p^{36} T^{46} + 17287941298996936 p^{38} T^{47} + 519330767244699 p^{40} T^{48} + 5094170773668 p^{42} T^{49} - 40799035126 p^{44} T^{50} - 1487543276 p^{46} T^{51} - 2865310 p^{48} T^{52} + 477588 p^{50} T^{53} + 12778 p^{52} T^{54} + 164 p^{54} T^{55} + p^{56} T^{56}$$
59 $$1 + 124 T - 2582 T^{2} - 676148 T^{3} + 18297186 T^{4} + 2494237964 T^{5} - 74034097270 T^{6} - 1291301782724 T^{7} + 13450026987579 T^{8} - 50932148532532808 T^{9} + 1191494304161568004 T^{10} +$$$$24\!\cdots\!84$$$$T^{11} -$$$$15\!\cdots\!72$$$$T^{12} -$$$$72\!\cdots\!80$$$$T^{13} +$$$$61\!\cdots\!16$$$$T^{14} -$$$$67\!\cdots\!40$$$$T^{15} -$$$$94\!\cdots\!19$$$$T^{16} +$$$$13\!\cdots\!36$$$$T^{17} -$$$$60\!\cdots\!58$$$$T^{18} -$$$$49\!\cdots\!72$$$$T^{19} +$$$$50\!\cdots\!34$$$$T^{20} +$$$$50\!\cdots\!24$$$$T^{21} -$$$$15\!\cdots\!82$$$$T^{22} +$$$$47\!\cdots\!00$$$$T^{23} +$$$$96\!\cdots\!83$$$$T^{24} -$$$$24\!\cdots\!52$$$$T^{25} +$$$$17\!\cdots\!76$$$$T^{26} +$$$$43\!\cdots\!08$$$$T^{27} -$$$$91\!\cdots\!12$$$$T^{28} +$$$$43\!\cdots\!08$$$$p^{2} T^{29} +$$$$17\!\cdots\!76$$$$p^{4} T^{30} -$$$$24\!\cdots\!52$$$$p^{6} T^{31} +$$$$96\!\cdots\!83$$$$p^{8} T^{32} +$$$$47\!\cdots\!00$$$$p^{10} T^{33} -$$$$15\!\cdots\!82$$$$p^{12} T^{34} +$$$$50\!\cdots\!24$$$$p^{14} T^{35} +$$$$50\!\cdots\!34$$$$p^{16} T^{36} -$$$$49\!\cdots\!72$$$$p^{18} T^{37} -$$$$60\!\cdots\!58$$$$p^{20} T^{38} +$$$$13\!\cdots\!36$$$$p^{22} T^{39} -$$$$94\!\cdots\!19$$$$p^{24} T^{40} -$$$$67\!\cdots\!40$$$$p^{26} T^{41} +$$$$61\!\cdots\!16$$$$p^{28} T^{42} -$$$$72\!\cdots\!80$$$$p^{30} T^{43} -$$$$15\!\cdots\!72$$$$p^{32} T^{44} +$$$$24\!\cdots\!84$$$$p^{34} T^{45} + 1191494304161568004 p^{36} T^{46} - 50932148532532808 p^{38} T^{47} + 13450026987579 p^{40} T^{48} - 1291301782724 p^{42} T^{49} - 74034097270 p^{44} T^{50} + 2494237964 p^{46} T^{51} + 18297186 p^{48} T^{52} - 676148 p^{50} T^{53} - 2582 p^{52} T^{54} + 124 p^{54} T^{55} + p^{56} T^{56}$$
61 $$1 + 68 T - 3382 T^{2} - 676620 T^{3} - 19307486 T^{4} + 550708916 T^{5} + 137721076522 T^{6} + 14290319529220 T^{7} + 879416281712219 T^{8} - 28458120472285176 T^{9} - 6519607438718241916 T^{10} -$$$$34\!\cdots\!92$$$$T^{11} +$$$$79\!\cdots\!92$$$$T^{12} +$$$$11\!\cdots\!56$$$$T^{13} +$$$$10\!\cdots\!72$$$$T^{14} +$$$$49\!\cdots\!56$$$$T^{15} -$$$$76\!\cdots\!35$$$$T^{16} -$$$$29\!\cdots\!84$$$$T^{17} -$$$$19\!\cdots\!74$$$$T^{18} -$$$$27\!\cdots\!96$$$$T^{19} +$$$$50\!\cdots\!66$$$$T^{20} +$$$$48\!\cdots\!24$$$$T^{21} +$$$$18\!\cdots\!06$$$$T^{22} -$$$$30\!\cdots\!80$$$$T^{23} -$$$$94\!\cdots\!53$$$$T^{24} -$$$$61\!\cdots\!20$$$$T^{25} -$$$$12\!\cdots\!76$$$$T^{26} +$$$$15\!\cdots\!48$$$$T^{27} +$$$$16\!\cdots\!80$$$$T^{28} +$$$$15\!\cdots\!48$$$$p^{2} T^{29} -$$$$12\!\cdots\!76$$$$p^{4} T^{30} -$$$$61\!\cdots\!20$$$$p^{6} T^{31} -$$$$94\!\cdots\!53$$$$p^{8} T^{32} -$$$$30\!\cdots\!80$$$$p^{10} T^{33} +$$$$18\!\cdots\!06$$$$p^{12} T^{34} +$$$$48\!\cdots\!24$$$$p^{14} T^{35} +$$$$50\!\cdots\!66$$$$p^{16} T^{36} -$$$$27\!\cdots\!96$$$$p^{18} T^{37} -$$$$19\!\cdots\!74$$$$p^{20} T^{38} -$$$$29\!\cdots\!84$$$$p^{22} T^{39} -$$$$76\!\cdots\!35$$$$p^{24} T^{40} +$$$$49\!\cdots\!56$$$$p^{26} T^{41} +$$$$10\!\cdots\!72$$$$p^{28} T^{42} +$$$$11\!\cdots\!56$$$$p^{30} T^{43} +$$$$79\!\cdots\!92$$$$p^{32} T^{44} -$$$$34\!\cdots\!92$$$$p^{34} T^{45} - 6519607438718241916 p^{36} T^{46} - 28458120472285176 p^{38} T^{47} + 879416281712219 p^{40} T^{48} + 14290319529220 p^{42} T^{49} + 137721076522 p^{44} T^{50} + 550708916 p^{46} T^{51} - 19307486 p^{48} T^{52} - 676620 p^{50} T^{53} - 3382 p^{52} T^{54} + 68 p^{54} T^{55} + p^{56} T^{56}$$
67 $$1 - 164 T + 1610 T^{2} + 539180 T^{3} + 50416674 T^{4} - 4728668884 T^{5} - 159879353814 T^{6} - 10686563025060 T^{7} + 2770502956998267 T^{8} - 36035490028130568 T^{9} + 2186996268372420676 T^{10} -$$$$74\!\cdots\!12$$$$T^{11} +$$$$13\!\cdots\!12$$$$T^{12} -$$$$55\!\cdots\!32$$$$T^{13} +$$$$28\!\cdots\!52$$$$T^{14} -$$$$15\!\cdots\!36$$$$T^{15} +$$$$37\!\cdots\!37$$$$T^{16} -$$$$68\!\cdots\!00$$$$T^{17} +$$$$60\!\cdots\!02$$$$T^{18} -$$$$23\!\cdots\!36$$$$T^{19} +$$$$12\!\cdots\!10$$$$T^{20} -$$$$14\!\cdots\!16$$$$T^{21} +$$$$93\!\cdots\!66$$$$T^{22} -$$$$57\!\cdots\!48$$$$T^{23} +$$$$92\!\cdots\!97$$$$p T^{24} -$$$$40\!\cdots\!52$$$$T^{25} +$$$$10\!\cdots\!00$$$$T^{26} -$$$$10\!\cdots\!64$$$$T^{27} +$$$$13\!\cdots\!16$$$$T^{28} -$$$$10\!\cdots\!64$$$$p^{2} T^{29} +$$$$10\!\cdots\!00$$$$p^{4} T^{30} -$$$$40\!\cdots\!52$$$$p^{6} T^{31} +$$$$92\!\cdots\!97$$$$p^{9} T^{32} -$$$$57\!\cdots\!48$$$$p^{10} T^{33} +$$$$93\!\cdots\!66$$$$p^{12} T^{34} -$$$$14\!\cdots\!16$$$$p^{14} T^{35} +$$$$12\!\cdots\!10$$$$p^{16} T^{36} -$$$$23\!\cdots\!36$$$$p^{18} T^{37} +$$$$60\!\cdots\!02$$$$p^{20} T^{38} -$$$$68\!\cdots\!00$$$$p^{22} T^{39} +$$$$37\!\cdots\!37$$$$p^{24} T^{40} -$$$$15\!\cdots\!36$$$$p^{26} T^{41} +$$$$28\!\cdots\!52$$$$p^{28} T^{42} -$$$$55\!\cdots\!32$$$$p^{30} T^{43} +$$$$13\!\cdots\!12$$$$p^{32} T^{44} -$$$$74\!\cdots\!12$$$$p^{34} T^{45} + 2186996268372420676 p^{36} T^{46} - 36035490028130568 p^{38} T^{47} + 2770502956998267 p^{40} T^{48} - 10686563025060 p^{42} T^{49} - 159879353814 p^{44} T^{50} - 4728668884 p^{46} T^{51} + 50416674 p^{48} T^{52} + 539180 p^{50} T^{53} + 1610 p^{52} T^{54} - 164 p^{54} T^{55} + p^{56} T^{56}$$
71 $$1 - 260 T + 33800 T^{2} - 3358028 T^{3} + 444069730 T^{4} - 58747785692 T^{5} + 5903043430312 T^{6} - 507612783055668 T^{7} + 50914532182356955 T^{8} - 5152210656262489336 T^{9} +$$$$42\!\cdots\!36$$$$T^{10} -$$$$31\!\cdots\!12$$$$T^{11} +$$$$27\!\cdots\!72$$$$T^{12} -$$$$23\!\cdots\!96$$$$T^{13} +$$$$15\!\cdots\!80$$$$T^{14} -$$$$10\!\cdots\!24$$$$T^{15} +$$$$76\!\cdots\!21$$$$T^{16} -$$$$53\!\cdots\!12$$$$T^{17} +$$$$25\!\cdots\!44$$$$T^{18} -$$$$10\!\cdots\!92$$$$T^{19} +$$$$65\!\cdots\!34$$$$T^{20} -$$$$14\!\cdots\!08$$$$T^{21} -$$$$37\!\cdots\!44$$$$T^{22} +$$$$43\!\cdots\!68$$$$T^{23} -$$$$30\!\cdots\!85$$$$T^{24} +$$$$31\!\cdots\!56$$$$T^{25} -$$$$31\!\cdots\!68$$$$T^{26} +$$$$21\!\cdots\!44$$$$T^{27} -$$$$13\!\cdots\!32$$$$T^{28} +$$$$21\!\cdots\!44$$$$p^{2} T^{29} -$$$$31\!\cdots\!68$$$$p^{4} T^{30} +$$$$31\!\cdots\!56$$$$p^{6} T^{31} -$$$$30\!\cdots\!85$$$$p^{8} T^{32} +$$$$43\!\cdots\!68$$$$p^{10} T^{33} -$$$$37\!\cdots\!44$$$$p^{12} T^{34} -$$$$14\!\cdots\!08$$$$p^{14} T^{35} +$$$$65\!\cdots\!34$$$$p^{16} T^{36} -$$$$10\!\cdots\!92$$$$p^{18} T^{37} +$$$$25\!\cdots\!44$$$$p^{20} T^{38} -$$$$53\!\cdots\!12$$$$p^{22} T^{39} +$$$$76\!\cdots\!21$$$$p^{24} T^{40} -$$$$10\!\cdots\!24$$$$p^{26} T^{41} +$$$$15\!\cdots\!80$$$$p^{28} T^{42} -$$$$23\!\cdots\!96$$$$p^{30} T^{43} +$$$$27\!\cdots\!72$$$$p^{32} T^{44} -$$$$31\!\cdots\!12$$$$p^{34} T^{45} +$$$$42\!\cdots\!36$$$$p^{36} T^{46} - 5152210656262489336 p^{38} T^{47} + 50914532182356955 p^{40} T^{48} - 507612783055668 p^{42} T^{49} + 5903043430312 p^{44} T^{50} - 58747785692 p^{46} T^{51} + 444069730 p^{48} T^{52} - 3358028 p^{50} T^{53} + 33800 p^{52} T^{54} - 260 p^{54} T^{55} + p^{56} T^{56}$$
73 $$1 + 4 T + 8 T^{2} + 162380 T^{3} - 20449662 T^{4} - 1481324836 T^{5} + 7421930152 T^{6} - 8318740383948 T^{7} + 728881382650971 T^{8} + 27208025429491064 T^{9} + 1141584982502320 T^{10} +$$$$10\!\cdots\!28$$$$T^{11} -$$$$39\!\cdots\!24$$$$T^{12} -$$$$64\!\cdots\!80$$$$T^{13} +$$$$29\!\cdots\!08$$$$T^{14} -$$$$68\!\cdots\!84$$$$T^{15} +$$$$17\!\cdots\!09$$$$T^{16} +$$$$35\!\cdots\!24$$$$T^{17} -$$$$13\!\cdots\!88$$$$T^{18} +$$$$16\!\cdots\!72$$$$T^{19} -$$$$38\!\cdots\!26$$$$T^{20} -$$$$36\!\cdots\!72$$$$T^{21} -$$$$16\!\cdots\!92$$$$T^{22} -$$$$10\!\cdots\!00$$$$T^{23} +$$$$15\!\cdots\!47$$$$T^{24} -$$$$16\!\cdots\!56$$$$p T^{25} -$$$$11\!\cdots\!00$$$$T^{26} +$$$$19\!\cdots\!24$$$$T^{27} -$$$$57\!\cdots\!36$$$$T^{28} +$$$$19\!\cdots\!24$$$$p^{2} T^{29} -$$$$11\!\cdots\!00$$$$p^{4} T^{30} -$$$$16\!\cdots\!56$$$$p^{7} T^{31} +$$$$15\!\cdots\!47$$$$p^{8} T^{32} -$$$$10\!\cdots\!00$$$$p^{10} T^{33} -$$$$16\!\cdots\!92$$$$p^{12} T^{34} -$$$$36\!\cdots\!72$$$$p^{14} T^{35} -$$$$38\!\cdots\!26$$$$p^{16} T^{36} +$$$$16\!\cdots\!72$$$$p^{18} T^{37} -$$$$13\!\cdots\!88$$$$p^{20} T^{38} +$$$$35\!\cdots\!24$$$$p^{22} T^{39} +$$$$17\!\cdots\!09$$$$p^{24} T^{40} -$$$$68\!\cdots\!84$$$$p^{26} T^{41} +$$$$29\!\cdots\!08$$$$p^{28} T^{42} -$$$$64\!\cdots\!80$$$$p^{30} T^{43} -$$$$39\!\cdots\!24$$$$p^{32} T^{44} +$$$$10\!\cdots\!28$$$$p^{34} T^{45} + 1141584982502320 p^{36} T^{46} + 27208025429491064 p^{38} T^{47} + 728881382650971 p^{40} T^{48} - 8318740383948 p^{42} T^{49} + 7421930152 p^{44} T^{50} - 1481324836 p^{46} T^{51} - 20449662 p^{48} T^{52} + 162380 p^{50} T^{53} + 8 p^{52} T^{54} + 4 p^{54} T^{55} + p^{56} T^{56}$$
79 $$( 1 - 260 T + 95642 T^{2} - 18771412 T^{3} + 4030082171 T^{4} - 637641691736 T^{5} + 101846137667940 T^{6} - 13492974421088792 T^{7} + 1748349708000644553 T^{8} -$$$$19\!\cdots\!64$$$$T^{9} +$$$$21\!\cdots\!54$$$$T^{10} -$$$$21\!\cdots\!24$$$$T^{11} +$$$$20\!\cdots\!83$$$$T^{12} -$$$$17\!\cdots\!76$$$$T^{13} +$$$$14\!\cdots\!72$$$$T^{14} -$$$$17\!\cdots\!76$$$$p^{2} T^{15} +$$$$20\!\cdots\!83$$$$p^{4} T^{16} -$$$$21\!\cdots\!24$$$$p^{6} T^{17} +$$$$21\!\cdots\!54$$$$p^{8} T^{18} -$$$$19\!\cdots\!64$$$$p^{10} T^{19} + 1748349708000644553 p^{12} T^{20} - 13492974421088792 p^{14} T^{21} + 101846137667940 p^{16} T^{22} - 637641691736 p^{18} T^{23} + 4030082171 p^{20} T^{24} - 18771412 p^{22} T^{25} + 95642 p^{24} T^{26} - 260 p^{26} T^{27} + p^{28} T^{28} )^{2}$$
83 $$1 - 484 T + 103498 T^{2} - 12003540 T^{3} + 639572258 T^{4} + 20142216748 T^{5} - 5586218114390 T^{6} + 290579540968988 T^{7} + 4467098840062459 T^{8} - 338973532787675656 T^{9} -$$$$11\!\cdots\!84$$$$T^{10} +$$$$61\!\cdots\!92$$$$T^{11} +$$$$13\!\cdots\!52$$$$T^{12} -$$$$16\!\cdots\!28$$$$T^{13} +$$$$17\!\cdots\!24$$$$T^{14} +$$$$85\!\cdots\!80$$$$T^{15} -$$$$73\!\cdots\!39$$$$T^{16} -$$$$11\!\cdots\!96$$$$T^{17} +$$$$12\!\cdots\!66$$$$T^{18} -$$$$37\!\cdots\!92$$$$T^{19} -$$$$32\!\cdots\!78$$$$T^{20} +$$$$24\!\cdots\!16$$$$T^{21} +$$$$13\!\cdots\!02$$$$T^{22} -$$$$24\!\cdots\!16$$$$T^{23} +$$$$73\!\cdots\!51$$$$T^{24} +$$$$72\!\cdots\!80$$$$T^{25} -$$$$45\!\cdots\!92$$$$T^{26} -$$$$55\!\cdots\!72$$$$T^{27} +$$$$89\!\cdots\!12$$$$T^{28} -$$$$55\!\cdots\!72$$$$p^{2} T^{29} -$$$$45\!\cdots\!92$$$$p^{4} T^{30} +$$$$72\!\cdots\!80$$$$p^{6} T^{31} +$$$$73\!\cdots\!51$$$$p^{8} T^{32} -$$$$24\!\cdots\!16$$$$p^{10} T^{33} +$$$$13\!\cdots\!02$$$$p^{12} T^{34} +$$$$24\!\cdots\!16$$$$p^{14} T^{35} -$$$$32\!\cdots\!78$$$$p^{16} T^{36} -$$$$37\!\cdots\!92$$$$p^{18} T^{37} +$$$$12\!\cdots\!66$$$$p^{20} T^{38} -$$$$11\!\cdots\!96$$$$p^{22} T^{39} -$$$$73\!\cdots\!39$$$$p^{24} T^{40} +$$$$85\!\cdots\!80$$$$p^{26} T^{41} +$$$$17\!\cdots\!24$$$$p^{28} T^{42} -$$$$16\!\cdots\!28$$$$p^{30} T^{43} +$$$$13\!\cdots\!52$$$$p^{32} T^{44} +$$$$61\!\cdots\!92$$$$p^{34} T^{45} -$$$$11\!\cdots\!84$$$$p^{36} T^{46} - 338973532787675656 p^{38} T^{47} + 4467098840062459 p^{40} T^{48} + 290579540968988 p^{42} T^{49} - 5586218114390 p^{44} T^{50} + 20142216748 p^{46} T^{51} + 639572258 p^{48} T^{52} - 12003540 p^{50} T^{53} + 103498 p^{52} T^{54} - 484 p^{54} T^{55} + p^{56} T^{56}$$
89 $$1 + 4 T + 8 T^{2} + 415308 T^{3} + 114976258 T^{4} - 956521892 T^{5} + 915668200 p T^{6} + 58702991801012 T^{7} + 10344826690180571 T^{8} - 269406900644620936 T^{9} + 14306973782434384816 T^{10} +$$$$27\!\cdots\!68$$$$T^{11} +$$$$90\!\cdots\!88$$$$T^{12} -$$$$57\!\cdots\!12$$$$T^{13} +$$$$61\!\cdots\!48$$$$T^{14} +$$$$18\!\cdots\!16$$$$T^{15} +$$$$82\!\cdots\!09$$$$T^{16} +$$$$33\!\cdots\!56$$$$T^{17} -$$$$11\!\cdots\!72$$$$T^{18} +$$$$17\!\cdots\!20$$$$T^{19} +$$$$69\!\cdots\!38$$$$T^{20} -$$$$86\!\cdots\!96$$$$T^{21} -$$$$18\!\cdots\!84$$$$T^{22} +$$$$96\!\cdots\!56$$$$T^{23} +$$$$45\!\cdots\!83$$$$T^{24} +$$$$71\!\cdots\!64$$$$T^{25} -$$$$11\!\cdots\!28$$$$T^{26} +$$$$37\!\cdots\!72$$$$T^{27} +$$$$26\!\cdots\!64$$$$T^{28} +$$$$37\!\cdots\!72$$$$p^{2} T^{29} -$$$$11\!\cdots\!28$$$$p^{4} T^{30} +$$$$71\!\cdots\!64$$$$p^{6} T^{31} +$$$$45\!\cdots\!83$$$$p^{8} T^{32} +$$$$96\!\cdots\!56$$$$p^{10} T^{33} -$$$$18\!\cdots\!84$$$$p^{12} T^{34} -$$$$86\!\cdots\!96$$$$p^{14} T^{35} +$$$$69\!\cdots\!38$$$$p^{16} T^{36} +$$$$17\!\cdots\!20$$$$p^{18} T^{37} -$$$$11\!\cdots\!72$$$$p^{20} T^{38} +$$$$33\!\cdots\!56$$$$p^{22} T^{39} +$$$$82\!\cdots\!09$$$$p^{24} T^{40} +$$$$18\!\cdots\!16$$$$p^{26} T^{41} +$$$$61\!\cdots\!48$$$$p^{28} T^{42} -$$$$57\!\cdots\!12$$$$p^{30} T^{43} +$$$$90\!\cdots\!88$$$$p^{32} T^{44} +$$$$27\!\cdots\!68$$$$p^{34} T^{45} + 14306973782434384816 p^{36} T^{46} - 269406900644620936 p^{38} T^{47} + 10344826690180571 p^{40} T^{48} + 58702991801012 p^{42} T^{49} + 915668200 p^{45} T^{50} - 956521892 p^{46} T^{51} + 114976258 p^{48} T^{52} + 415308 p^{50} T^{53} + 8 p^{52} T^{54} + 4 p^{54} T^{55} + p^{56} T^{56}$$
97 $$( 1 + 4 T + 78474 T^{2} + 78868 T^{3} + 3115261595 T^{4} - 9513593064 T^{5} + 82085510621700 T^{6} - 594744065858344 T^{7} + 1601504954000400553 T^{8} - 17354013974432109428 T^{9} +$$$$24\!\cdots\!38$$$$T^{10} -$$$$32\!\cdots\!56$$$$T^{11} +$$$$30\!\cdots\!19$$$$T^{12} -$$$$41\!\cdots\!40$$$$T^{13} +$$$$31\!\cdots\!64$$$$T^{14} -$$$$41\!\cdots\!40$$$$p^{2} T^{15} +$$$$30\!\cdots\!19$$$$p^{4} T^{16} -$$$$32\!\cdots\!56$$$$p^{6} T^{17} +$$$$24\!\cdots\!38$$$$p^{8} T^{18} - 17354013974432109428 p^{10} T^{19} + 1601504954000400553 p^{12} T^{20} - 594744065858344 p^{14} T^{21} + 82085510621700 p^{16} T^{22} - 9513593064 p^{18} T^{23} + 3115261595 p^{20} T^{24} + 78868 p^{22} T^{25} + 78474 p^{24} T^{26} + 4 p^{26} T^{27} + p^{28} T^{28} )^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{56} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$