Properties

 Degree 68 Conductor $2^{34} \cdot 19^{34} \cdot 211^{34}$ Sign $1$ Motivic weight 1 Primitive no Self-dual yes Analytic rank 34

Origins of factors

Dirichlet series

 L(s)  = 1 − 34·2-s − 10·3-s + 595·4-s + 7·5-s + 340·6-s − 6·7-s − 7.14e3·8-s + 18·9-s − 238·10-s + 4·11-s − 5.95e3·12-s − 5·13-s + 204·14-s − 70·15-s + 6.60e4·16-s + 8·17-s − 612·18-s − 34·19-s + 4.16e3·20-s + 60·21-s − 136·22-s − 24·23-s + 7.14e4·24-s − 58·25-s + 170·26-s + 153·27-s − 3.57e3·28-s + ⋯
 L(s)  = 1 − 24.0·2-s − 5.77·3-s + 297.5·4-s + 3.13·5-s + 138.·6-s − 2.26·7-s − 2.52e3·8-s + 6·9-s − 75.2·10-s + 1.20·11-s − 1.71e3·12-s − 1.38·13-s + 54.5·14-s − 18.0·15-s + 1.65e4·16-s + 1.94·17-s − 144.·18-s − 7.80·19-s + 931.·20-s + 13.0·21-s − 28.9·22-s − 5.00·23-s + 1.45e4·24-s − 11.5·25-s + 33.3·26-s + 29.4·27-s − 674.·28-s + ⋯

Functional equation

\begin{aligned} \Lambda(s)=\mathstrut &\left(2^{34} \cdot 19^{34} \cdot 211^{34}\right)^{s/2} \, \Gamma_{\C}(s)^{34} \, L(s)\cr =\mathstrut & \,\Lambda(2-s) \end{aligned}
\begin{aligned} \Lambda(s)=\mathstrut &\left(2^{34} \cdot 19^{34} \cdot 211^{34}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{34} \, L(s)\cr =\mathstrut & \,\Lambda(1-s) \end{aligned}

Invariants

 $$d$$ = $$68$$ $$N$$ = $$2^{34} \cdot 19^{34} \cdot 211^{34}$$ $$\varepsilon$$ = $1$ motivic weight = $$1$$ character : induced by $\chi_{8018} (1, \cdot )$ primitive : no self-dual : yes analytic rank = 34 Selberg data = $(68,\ 2^{34} \cdot 19^{34} \cdot 211^{34} ,\ ( \ : [1/2]^{34} ),\ 1 )$ $L(1)$ $=$ $0$ $L(\frac12)$ $=$ $0$ $L(\frac{3}{2})$ not available $L(1)$ not available

Euler product

$L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1}$ where, for $p \notin \{2,\;19,\;211\}$, $$F_p$$ is a polynomial of degree 68. If $p \in \{2,\;19,\;211\}$, then $F_p$ is a polynomial of degree at most 67.
$p$$F_p$
bad2 $$( 1 + T )^{34}$$
19 $$( 1 + T )^{34}$$
211 $$( 1 + T )^{34}$$
good3 $$1 + 10 T + 82 T^{2} + 487 T^{3} + 853 p T^{4} + 1286 p^{2} T^{5} + 48037 T^{6} + 181525 T^{7} + 642676 T^{8} + 2127136 T^{9} + 6683647 T^{10} + 19930270 T^{11} + 56929315 T^{12} + 17313305 p^{2} T^{13} + 137066779 p T^{14} + 1046766430 T^{15} + 2581784743 T^{16} + 6172981372 T^{17} + 14355446216 T^{18} + 32486570980 T^{19} + 71730732818 T^{20} + 154605373375 T^{21} + 326003235623 T^{22} + 224268182621 p T^{23} + 453879088738 p T^{24} + 2703458220415 T^{25} + 5274863338301 T^{26} + 10118055156211 T^{27} + 19109468246006 T^{28} + 35545727599784 T^{29} + 65207842722815 T^{30} + 117987431764814 T^{31} + 210801152103212 T^{32} + 371837088313921 T^{33} + 216021664106884 p T^{34} + 371837088313921 p T^{35} + 210801152103212 p^{2} T^{36} + 117987431764814 p^{3} T^{37} + 65207842722815 p^{4} T^{38} + 35545727599784 p^{5} T^{39} + 19109468246006 p^{6} T^{40} + 10118055156211 p^{7} T^{41} + 5274863338301 p^{8} T^{42} + 2703458220415 p^{9} T^{43} + 453879088738 p^{11} T^{44} + 224268182621 p^{12} T^{45} + 326003235623 p^{12} T^{46} + 154605373375 p^{13} T^{47} + 71730732818 p^{14} T^{48} + 32486570980 p^{15} T^{49} + 14355446216 p^{16} T^{50} + 6172981372 p^{17} T^{51} + 2581784743 p^{18} T^{52} + 1046766430 p^{19} T^{53} + 137066779 p^{21} T^{54} + 17313305 p^{23} T^{55} + 56929315 p^{22} T^{56} + 19930270 p^{23} T^{57} + 6683647 p^{24} T^{58} + 2127136 p^{25} T^{59} + 642676 p^{26} T^{60} + 181525 p^{27} T^{61} + 48037 p^{28} T^{62} + 1286 p^{31} T^{63} + 853 p^{31} T^{64} + 487 p^{31} T^{65} + 82 p^{32} T^{66} + 10 p^{33} T^{67} + p^{34} T^{68}$$
5 $$1 - 7 T + 107 T^{2} - 643 T^{3} + 1119 p T^{4} - 29556 T^{5} + 191348 T^{6} - 904983 T^{7} + 4826551 T^{8} - 20732552 T^{9} + 95920109 T^{10} - 378497069 T^{11} + 313182557 p T^{12} - 5728281623 T^{13} + 21612191397 T^{14} - 73841011911 T^{15} + 257542973596 T^{16} - 165392327736 p T^{17} + 538624458353 p T^{18} - 8170030959593 T^{19} + 25036326709424 T^{20} - 72090219143213 T^{21} + 209180002868354 T^{22} - 574066317601024 T^{23} + 1585609834204597 T^{24} - 4163071969846966 T^{25} + 10995690901727203 T^{26} - 27714658416281748 T^{27} + 70276732657943439 T^{28} - 6822545753066782 p^{2} T^{29} + 416600302099888861 T^{30} - 976011979463203328 T^{31} + 2302043487546885373 T^{32} - 5214833252494133128 T^{33} + 2379171478291106374 p T^{34} - 5214833252494133128 p T^{35} + 2302043487546885373 p^{2} T^{36} - 976011979463203328 p^{3} T^{37} + 416600302099888861 p^{4} T^{38} - 6822545753066782 p^{7} T^{39} + 70276732657943439 p^{6} T^{40} - 27714658416281748 p^{7} T^{41} + 10995690901727203 p^{8} T^{42} - 4163071969846966 p^{9} T^{43} + 1585609834204597 p^{10} T^{44} - 574066317601024 p^{11} T^{45} + 209180002868354 p^{12} T^{46} - 72090219143213 p^{13} T^{47} + 25036326709424 p^{14} T^{48} - 8170030959593 p^{15} T^{49} + 538624458353 p^{17} T^{50} - 165392327736 p^{18} T^{51} + 257542973596 p^{18} T^{52} - 73841011911 p^{19} T^{53} + 21612191397 p^{20} T^{54} - 5728281623 p^{21} T^{55} + 313182557 p^{23} T^{56} - 378497069 p^{23} T^{57} + 95920109 p^{24} T^{58} - 20732552 p^{25} T^{59} + 4826551 p^{26} T^{60} - 904983 p^{27} T^{61} + 191348 p^{28} T^{62} - 29556 p^{29} T^{63} + 1119 p^{31} T^{64} - 643 p^{31} T^{65} + 107 p^{32} T^{66} - 7 p^{33} T^{67} + p^{34} T^{68}$$
7 $$1 + 6 T + 18 p T^{2} + 97 p T^{3} + 7709 T^{4} + 5356 p T^{5} + 304511 T^{6} + 1343665 T^{7} + 8719367 T^{8} + 35090756 T^{9} + 192837437 T^{10} + 711352382 T^{11} + 3431376555 T^{12} + 11659462091 T^{13} + 50620804133 T^{14} + 159276183022 T^{15} + 634826879943 T^{16} + 1861414989660 T^{17} + 6930648894882 T^{18} + 2727262654600 p T^{19} + 9642050139968 p T^{20} + 176373532654043 T^{21} + 600858248956353 T^{22} + 214974682627553 p T^{23} + 4996305596176950 T^{24} + 12091234048308788 T^{25} + 39396837610793597 T^{26} + 92571873524887426 T^{27} + 297213022104951657 T^{28} + 680278586173772436 T^{29} + 2162025467610646840 T^{30} + 4847024227148470238 T^{31} + 15343404290279844686 T^{32} + 34005974493977288307 T^{33} +$$$$10\!\cdots\!82$$$$T^{34} + 34005974493977288307 p T^{35} + 15343404290279844686 p^{2} T^{36} + 4847024227148470238 p^{3} T^{37} + 2162025467610646840 p^{4} T^{38} + 680278586173772436 p^{5} T^{39} + 297213022104951657 p^{6} T^{40} + 92571873524887426 p^{7} T^{41} + 39396837610793597 p^{8} T^{42} + 12091234048308788 p^{9} T^{43} + 4996305596176950 p^{10} T^{44} + 214974682627553 p^{12} T^{45} + 600858248956353 p^{12} T^{46} + 176373532654043 p^{13} T^{47} + 9642050139968 p^{15} T^{48} + 2727262654600 p^{16} T^{49} + 6930648894882 p^{16} T^{50} + 1861414989660 p^{17} T^{51} + 634826879943 p^{18} T^{52} + 159276183022 p^{19} T^{53} + 50620804133 p^{20} T^{54} + 11659462091 p^{21} T^{55} + 3431376555 p^{22} T^{56} + 711352382 p^{23} T^{57} + 192837437 p^{24} T^{58} + 35090756 p^{25} T^{59} + 8719367 p^{26} T^{60} + 1343665 p^{27} T^{61} + 304511 p^{28} T^{62} + 5356 p^{30} T^{63} + 7709 p^{30} T^{64} + 97 p^{32} T^{65} + 18 p^{33} T^{66} + 6 p^{33} T^{67} + p^{34} T^{68}$$
11 $$1 - 4 T + 173 T^{2} - 780 T^{3} + 15711 T^{4} - 75215 T^{5} + 989477 T^{6} - 4836973 T^{7} + 48142010 T^{8} - 234455707 T^{9} + 1914060466 T^{10} - 9148872230 T^{11} + 64360896700 T^{12} - 299298760982 T^{13} + 1873451890617 T^{14} - 8434527415035 T^{15} + 48012338577996 T^{16} - 208728962761403 T^{17} + 1097241332380064 T^{18} - 4600615325823744 T^{19} + 22583852851216697 T^{20} - 91285614651163612 T^{21} + 421920371621658205 T^{22} - 1643999431706552429 T^{23} + 7199087086116331493 T^{24} - 2458465976613478446 p T^{25} +$$$$11\!\cdots\!93$$$$T^{26} -$$$$40\!\cdots\!06$$$$T^{27} +$$$$14\!\cdots\!31$$$$p T^{28} -$$$$56\!\cdots\!75$$$$T^{29} +$$$$21\!\cdots\!93$$$$T^{30} -$$$$72\!\cdots\!71$$$$T^{31} +$$$$26\!\cdots\!75$$$$T^{32} -$$$$86\!\cdots\!08$$$$T^{33} +$$$$30\!\cdots\!56$$$$T^{34} -$$$$86\!\cdots\!08$$$$p T^{35} +$$$$26\!\cdots\!75$$$$p^{2} T^{36} -$$$$72\!\cdots\!71$$$$p^{3} T^{37} +$$$$21\!\cdots\!93$$$$p^{4} T^{38} -$$$$56\!\cdots\!75$$$$p^{5} T^{39} +$$$$14\!\cdots\!31$$$$p^{7} T^{40} -$$$$40\!\cdots\!06$$$$p^{7} T^{41} +$$$$11\!\cdots\!93$$$$p^{8} T^{42} - 2458465976613478446 p^{10} T^{43} + 7199087086116331493 p^{10} T^{44} - 1643999431706552429 p^{11} T^{45} + 421920371621658205 p^{12} T^{46} - 91285614651163612 p^{13} T^{47} + 22583852851216697 p^{14} T^{48} - 4600615325823744 p^{15} T^{49} + 1097241332380064 p^{16} T^{50} - 208728962761403 p^{17} T^{51} + 48012338577996 p^{18} T^{52} - 8434527415035 p^{19} T^{53} + 1873451890617 p^{20} T^{54} - 299298760982 p^{21} T^{55} + 64360896700 p^{22} T^{56} - 9148872230 p^{23} T^{57} + 1914060466 p^{24} T^{58} - 234455707 p^{25} T^{59} + 48142010 p^{26} T^{60} - 4836973 p^{27} T^{61} + 989477 p^{28} T^{62} - 75215 p^{29} T^{63} + 15711 p^{30} T^{64} - 780 p^{31} T^{65} + 173 p^{32} T^{66} - 4 p^{33} T^{67} + p^{34} T^{68}$$
13 $$1 + 5 T + 240 T^{2} + 1162 T^{3} + 2218 p T^{4} + 801 p^{2} T^{5} + 2314735 T^{6} + 10539090 T^{7} + 10748436 p T^{8} + 47440843 p T^{9} + 6763959260 T^{10} + 28924361631 T^{11} + 273358253343 T^{12} + 1131959491155 T^{13} + 729218316893 p T^{14} + 37999726496988 T^{15} + 287737901156926 T^{16} + 85866089972458 p T^{17} + 7758128229528035 T^{18} + 29124846447152056 T^{19} + 187945488539712775 T^{20} + 682722758636803426 T^{21} + 4127550328218651223 T^{22} + 14506674859630802572 T^{23} + 82752940023530637905 T^{24} +$$$$28\!\cdots\!88$$$$T^{25} +$$$$15\!\cdots\!81$$$$T^{26} +$$$$50\!\cdots\!87$$$$T^{27} +$$$$25\!\cdots\!54$$$$T^{28} +$$$$82\!\cdots\!86$$$$T^{29} +$$$$40\!\cdots\!93$$$$T^{30} +$$$$12\!\cdots\!02$$$$T^{31} +$$$$58\!\cdots\!54$$$$T^{32} +$$$$17\!\cdots\!58$$$$T^{33} +$$$$79\!\cdots\!44$$$$T^{34} +$$$$17\!\cdots\!58$$$$p T^{35} +$$$$58\!\cdots\!54$$$$p^{2} T^{36} +$$$$12\!\cdots\!02$$$$p^{3} T^{37} +$$$$40\!\cdots\!93$$$$p^{4} T^{38} +$$$$82\!\cdots\!86$$$$p^{5} T^{39} +$$$$25\!\cdots\!54$$$$p^{6} T^{40} +$$$$50\!\cdots\!87$$$$p^{7} T^{41} +$$$$15\!\cdots\!81$$$$p^{8} T^{42} +$$$$28\!\cdots\!88$$$$p^{9} T^{43} + 82752940023530637905 p^{10} T^{44} + 14506674859630802572 p^{11} T^{45} + 4127550328218651223 p^{12} T^{46} + 682722758636803426 p^{13} T^{47} + 187945488539712775 p^{14} T^{48} + 29124846447152056 p^{15} T^{49} + 7758128229528035 p^{16} T^{50} + 85866089972458 p^{18} T^{51} + 287737901156926 p^{18} T^{52} + 37999726496988 p^{19} T^{53} + 729218316893 p^{21} T^{54} + 1131959491155 p^{21} T^{55} + 273358253343 p^{22} T^{56} + 28924361631 p^{23} T^{57} + 6763959260 p^{24} T^{58} + 47440843 p^{26} T^{59} + 10748436 p^{27} T^{60} + 10539090 p^{27} T^{61} + 2314735 p^{28} T^{62} + 801 p^{31} T^{63} + 2218 p^{31} T^{64} + 1162 p^{31} T^{65} + 240 p^{32} T^{66} + 5 p^{33} T^{67} + p^{34} T^{68}$$
17 $$1 - 8 T + 354 T^{2} - 2461 T^{3} + 60032 T^{4} - 368300 T^{5} + 6546036 T^{6} - 35851902 T^{7} + 519419760 T^{8} - 2563379056 T^{9} + 32173875518 T^{10} - 144260009773 T^{11} + 1629621067452 T^{12} - 6692066007236 T^{13} + 69795751321134 T^{14} - 264687181343889 T^{15} + 152529095419954 p T^{16} - 9160759178886876 T^{17} + 85238113936478663 T^{18} - 283083937782056298 T^{19} + 148136035266782677 p T^{20} - 7932475551865243230 T^{21} + 67696651488661394576 T^{22} -$$$$20\!\cdots\!74$$$$T^{23} +$$$$16\!\cdots\!03$$$$T^{24} -$$$$48\!\cdots\!75$$$$T^{25} +$$$$38\!\cdots\!46$$$$T^{26} -$$$$10\!\cdots\!01$$$$T^{27} +$$$$81\!\cdots\!11$$$$T^{28} -$$$$22\!\cdots\!31$$$$T^{29} +$$$$16\!\cdots\!82$$$$T^{30} -$$$$42\!\cdots\!94$$$$T^{31} +$$$$30\!\cdots\!62$$$$T^{32} -$$$$77\!\cdots\!52$$$$T^{33} +$$$$52\!\cdots\!82$$$$T^{34} -$$$$77\!\cdots\!52$$$$p T^{35} +$$$$30\!\cdots\!62$$$$p^{2} T^{36} -$$$$42\!\cdots\!94$$$$p^{3} T^{37} +$$$$16\!\cdots\!82$$$$p^{4} T^{38} -$$$$22\!\cdots\!31$$$$p^{5} T^{39} +$$$$81\!\cdots\!11$$$$p^{6} T^{40} -$$$$10\!\cdots\!01$$$$p^{7} T^{41} +$$$$38\!\cdots\!46$$$$p^{8} T^{42} -$$$$48\!\cdots\!75$$$$p^{9} T^{43} +$$$$16\!\cdots\!03$$$$p^{10} T^{44} -$$$$20\!\cdots\!74$$$$p^{11} T^{45} + 67696651488661394576 p^{12} T^{46} - 7932475551865243230 p^{13} T^{47} + 148136035266782677 p^{15} T^{48} - 283083937782056298 p^{15} T^{49} + 85238113936478663 p^{16} T^{50} - 9160759178886876 p^{17} T^{51} + 152529095419954 p^{19} T^{52} - 264687181343889 p^{19} T^{53} + 69795751321134 p^{20} T^{54} - 6692066007236 p^{21} T^{55} + 1629621067452 p^{22} T^{56} - 144260009773 p^{23} T^{57} + 32173875518 p^{24} T^{58} - 2563379056 p^{25} T^{59} + 519419760 p^{26} T^{60} - 35851902 p^{27} T^{61} + 6546036 p^{28} T^{62} - 368300 p^{29} T^{63} + 60032 p^{30} T^{64} - 2461 p^{31} T^{65} + 354 p^{32} T^{66} - 8 p^{33} T^{67} + p^{34} T^{68}$$
23 $$1 + 24 T + 29 p T^{2} + 11574 T^{3} + 198280 T^{4} + 2763010 T^{5} + 36814330 T^{6} + 435812065 T^{7} + 4911114061 T^{8} + 51124491211 T^{9} + 508035937550 T^{10} + 4758688369371 T^{11} + 42738142137641 T^{12} + 366070590584010 T^{13} + 3019428100976322 T^{14} + 23930404815264367 T^{15} + 183319643772231413 T^{16} + 1356285843724494119 T^{17} + 9728995643729749784 T^{18} + 67649309273036401441 T^{19} +$$$$45\!\cdots\!06$$$$T^{20} +$$$$30\!\cdots\!17$$$$T^{21} +$$$$19\!\cdots\!59$$$$T^{22} +$$$$11\!\cdots\!30$$$$T^{23} +$$$$72\!\cdots\!78$$$$T^{24} +$$$$43\!\cdots\!86$$$$T^{25} +$$$$25\!\cdots\!69$$$$T^{26} +$$$$14\!\cdots\!37$$$$T^{27} +$$$$78\!\cdots\!80$$$$T^{28} +$$$$42\!\cdots\!16$$$$T^{29} +$$$$22\!\cdots\!82$$$$T^{30} +$$$$11\!\cdots\!78$$$$T^{31} +$$$$58\!\cdots\!89$$$$T^{32} +$$$$28\!\cdots\!38$$$$T^{33} +$$$$14\!\cdots\!96$$$$T^{34} +$$$$28\!\cdots\!38$$$$p T^{35} +$$$$58\!\cdots\!89$$$$p^{2} T^{36} +$$$$11\!\cdots\!78$$$$p^{3} T^{37} +$$$$22\!\cdots\!82$$$$p^{4} T^{38} +$$$$42\!\cdots\!16$$$$p^{5} T^{39} +$$$$78\!\cdots\!80$$$$p^{6} T^{40} +$$$$14\!\cdots\!37$$$$p^{7} T^{41} +$$$$25\!\cdots\!69$$$$p^{8} T^{42} +$$$$43\!\cdots\!86$$$$p^{9} T^{43} +$$$$72\!\cdots\!78$$$$p^{10} T^{44} +$$$$11\!\cdots\!30$$$$p^{11} T^{45} +$$$$19\!\cdots\!59$$$$p^{12} T^{46} +$$$$30\!\cdots\!17$$$$p^{13} T^{47} +$$$$45\!\cdots\!06$$$$p^{14} T^{48} + 67649309273036401441 p^{15} T^{49} + 9728995643729749784 p^{16} T^{50} + 1356285843724494119 p^{17} T^{51} + 183319643772231413 p^{18} T^{52} + 23930404815264367 p^{19} T^{53} + 3019428100976322 p^{20} T^{54} + 366070590584010 p^{21} T^{55} + 42738142137641 p^{22} T^{56} + 4758688369371 p^{23} T^{57} + 508035937550 p^{24} T^{58} + 51124491211 p^{25} T^{59} + 4911114061 p^{26} T^{60} + 435812065 p^{27} T^{61} + 36814330 p^{28} T^{62} + 2763010 p^{29} T^{63} + 198280 p^{30} T^{64} + 11574 p^{31} T^{65} + 29 p^{33} T^{66} + 24 p^{33} T^{67} + p^{34} T^{68}$$
29 $$1 - 24 T + 815 T^{2} - 14669 T^{3} + 294976 T^{4} - 4320746 T^{5} + 65657806 T^{6} - 819852202 T^{7} + 10305349742 T^{8} - 113007447017 T^{9} + 1230544591466 T^{10} - 12099049999501 T^{11} + 117376770087222 T^{12} - 1050943816906082 T^{13} + 9258906249495044 T^{14} - 76426948816368465 T^{15} + 620283177471055541 T^{16} - 4769289618875302490 T^{17} + 36072059139816430615 T^{18} -$$$$26\!\cdots\!27$$$$T^{19} + 63989771459249242951 p T^{20} -$$$$12\!\cdots\!28$$$$T^{21} +$$$$85\!\cdots\!70$$$$T^{22} -$$$$56\!\cdots\!20$$$$T^{23} +$$$$36\!\cdots\!18$$$$T^{24} -$$$$22\!\cdots\!50$$$$T^{25} +$$$$14\!\cdots\!07$$$$T^{26} -$$$$85\!\cdots\!26$$$$T^{27} +$$$$51\!\cdots\!33$$$$T^{28} -$$$$30\!\cdots\!89$$$$T^{29} +$$$$17\!\cdots\!22$$$$T^{30} -$$$$99\!\cdots\!61$$$$T^{31} +$$$$55\!\cdots\!88$$$$T^{32} -$$$$30\!\cdots\!09$$$$T^{33} +$$$$16\!\cdots\!25$$$$T^{34} -$$$$30\!\cdots\!09$$$$p T^{35} +$$$$55\!\cdots\!88$$$$p^{2} T^{36} -$$$$99\!\cdots\!61$$$$p^{3} T^{37} +$$$$17\!\cdots\!22$$$$p^{4} T^{38} -$$$$30\!\cdots\!89$$$$p^{5} T^{39} +$$$$51\!\cdots\!33$$$$p^{6} T^{40} -$$$$85\!\cdots\!26$$$$p^{7} T^{41} +$$$$14\!\cdots\!07$$$$p^{8} T^{42} -$$$$22\!\cdots\!50$$$$p^{9} T^{43} +$$$$36\!\cdots\!18$$$$p^{10} T^{44} -$$$$56\!\cdots\!20$$$$p^{11} T^{45} +$$$$85\!\cdots\!70$$$$p^{12} T^{46} -$$$$12\!\cdots\!28$$$$p^{13} T^{47} + 63989771459249242951 p^{15} T^{48} -$$$$26\!\cdots\!27$$$$p^{15} T^{49} + 36072059139816430615 p^{16} T^{50} - 4769289618875302490 p^{17} T^{51} + 620283177471055541 p^{18} T^{52} - 76426948816368465 p^{19} T^{53} + 9258906249495044 p^{20} T^{54} - 1050943816906082 p^{21} T^{55} + 117376770087222 p^{22} T^{56} - 12099049999501 p^{23} T^{57} + 1230544591466 p^{24} T^{58} - 113007447017 p^{25} T^{59} + 10305349742 p^{26} T^{60} - 819852202 p^{27} T^{61} + 65657806 p^{28} T^{62} - 4320746 p^{29} T^{63} + 294976 p^{30} T^{64} - 14669 p^{31} T^{65} + 815 p^{32} T^{66} - 24 p^{33} T^{67} + p^{34} T^{68}$$
31 $$1 + 28 T + 883 T^{2} + 17551 T^{3} + 344382 T^{4} + 5484507 T^{5} + 84047043 T^{6} + 1139166076 T^{7} + 14798916254 T^{8} + 176864564562 T^{9} + 65464579856 p T^{10} + 21880874563991 T^{11} + 227201659896799 T^{12} + 2245132203334675 T^{13} + 21433317314355048 T^{14} + 196330039996766353 T^{15} + 1742328350992243916 T^{16} + 14919747831276266306 T^{17} +$$$$12\!\cdots\!85$$$$T^{18} +$$$$99\!\cdots\!85$$$$T^{19} +$$$$78\!\cdots\!48$$$$T^{20} +$$$$59\!\cdots\!57$$$$T^{21} +$$$$44\!\cdots\!68$$$$T^{22} +$$$$32\!\cdots\!60$$$$T^{23} +$$$$22\!\cdots\!86$$$$T^{24} +$$$$15\!\cdots\!78$$$$T^{25} +$$$$10\!\cdots\!88$$$$T^{26} +$$$$68\!\cdots\!52$$$$T^{27} +$$$$44\!\cdots\!59$$$$T^{28} +$$$$27\!\cdots\!52$$$$T^{29} +$$$$17\!\cdots\!85$$$$T^{30} +$$$$10\!\cdots\!38$$$$T^{31} +$$$$60\!\cdots\!87$$$$T^{32} +$$$$34\!\cdots\!61$$$$T^{33} +$$$$19\!\cdots\!72$$$$T^{34} +$$$$34\!\cdots\!61$$$$p T^{35} +$$$$60\!\cdots\!87$$$$p^{2} T^{36} +$$$$10\!\cdots\!38$$$$p^{3} T^{37} +$$$$17\!\cdots\!85$$$$p^{4} T^{38} +$$$$27\!\cdots\!52$$$$p^{5} T^{39} +$$$$44\!\cdots\!59$$$$p^{6} T^{40} +$$$$68\!\cdots\!52$$$$p^{7} T^{41} +$$$$10\!\cdots\!88$$$$p^{8} T^{42} +$$$$15\!\cdots\!78$$$$p^{9} T^{43} +$$$$22\!\cdots\!86$$$$p^{10} T^{44} +$$$$32\!\cdots\!60$$$$p^{11} T^{45} +$$$$44\!\cdots\!68$$$$p^{12} T^{46} +$$$$59\!\cdots\!57$$$$p^{13} T^{47} +$$$$78\!\cdots\!48$$$$p^{14} T^{48} +$$$$99\!\cdots\!85$$$$p^{15} T^{49} +$$$$12\!\cdots\!85$$$$p^{16} T^{50} + 14919747831276266306 p^{17} T^{51} + 1742328350992243916 p^{18} T^{52} + 196330039996766353 p^{19} T^{53} + 21433317314355048 p^{20} T^{54} + 2245132203334675 p^{21} T^{55} + 227201659896799 p^{22} T^{56} + 21880874563991 p^{23} T^{57} + 65464579856 p^{25} T^{58} + 176864564562 p^{25} T^{59} + 14798916254 p^{26} T^{60} + 1139166076 p^{27} T^{61} + 84047043 p^{28} T^{62} + 5484507 p^{29} T^{63} + 344382 p^{30} T^{64} + 17551 p^{31} T^{65} + 883 p^{32} T^{66} + 28 p^{33} T^{67} + p^{34} T^{68}$$
37 $$1 + 36 T + 1238 T^{2} + 29180 T^{3} + 634190 T^{4} + 11616640 T^{5} + 198041311 T^{6} + 3042206104 T^{7} + 43997156425 T^{8} + 591062837873 T^{9} + 7545449788905 T^{10} + 90980807487560 T^{11} + 1049869655221494 T^{12} + 11560950271660030 T^{13} + 122491848268928236 T^{14} + 1246994151809735600 T^{15} + 12264837311503236619 T^{16} +$$$$11\!\cdots\!99$$$$T^{17} +$$$$10\!\cdots\!54$$$$T^{18} +$$$$95\!\cdots\!77$$$$T^{19} +$$$$82\!\cdots\!97$$$$T^{20} +$$$$69\!\cdots\!30$$$$T^{21} +$$$$57\!\cdots\!85$$$$T^{22} +$$$$45\!\cdots\!98$$$$T^{23} +$$$$35\!\cdots\!89$$$$T^{24} +$$$$26\!\cdots\!47$$$$T^{25} +$$$$19\!\cdots\!84$$$$T^{26} +$$$$14\!\cdots\!01$$$$T^{27} +$$$$10\!\cdots\!10$$$$T^{28} +$$$$69\!\cdots\!79$$$$T^{29} +$$$$46\!\cdots\!60$$$$T^{30} +$$$$30\!\cdots\!59$$$$T^{31} +$$$$19\!\cdots\!43$$$$T^{32} +$$$$12\!\cdots\!55$$$$T^{33} +$$$$76\!\cdots\!46$$$$T^{34} +$$$$12\!\cdots\!55$$$$p T^{35} +$$$$19\!\cdots\!43$$$$p^{2} T^{36} +$$$$30\!\cdots\!59$$$$p^{3} T^{37} +$$$$46\!\cdots\!60$$$$p^{4} T^{38} +$$$$69\!\cdots\!79$$$$p^{5} T^{39} +$$$$10\!\cdots\!10$$$$p^{6} T^{40} +$$$$14\!\cdots\!01$$$$p^{7} T^{41} +$$$$19\!\cdots\!84$$$$p^{8} T^{42} +$$$$26\!\cdots\!47$$$$p^{9} T^{43} +$$$$35\!\cdots\!89$$$$p^{10} T^{44} +$$$$45\!\cdots\!98$$$$p^{11} T^{45} +$$$$57\!\cdots\!85$$$$p^{12} T^{46} +$$$$69\!\cdots\!30$$$$p^{13} T^{47} +$$$$82\!\cdots\!97$$$$p^{14} T^{48} +$$$$95\!\cdots\!77$$$$p^{15} T^{49} +$$$$10\!\cdots\!54$$$$p^{16} T^{50} +$$$$11\!\cdots\!99$$$$p^{17} T^{51} + 12264837311503236619 p^{18} T^{52} + 1246994151809735600 p^{19} T^{53} + 122491848268928236 p^{20} T^{54} + 11560950271660030 p^{21} T^{55} + 1049869655221494 p^{22} T^{56} + 90980807487560 p^{23} T^{57} + 7545449788905 p^{24} T^{58} + 591062837873 p^{25} T^{59} + 43997156425 p^{26} T^{60} + 3042206104 p^{27} T^{61} + 198041311 p^{28} T^{62} + 11616640 p^{29} T^{63} + 634190 p^{30} T^{64} + 29180 p^{31} T^{65} + 1238 p^{32} T^{66} + 36 p^{33} T^{67} + p^{34} T^{68}$$
41 $$1 - 9 T + 769 T^{2} - 6487 T^{3} + 286828 T^{4} - 2270638 T^{5} + 69323008 T^{6} - 515441351 T^{7} + 12247835435 T^{8} - 85586432053 T^{9} + 1693715300257 T^{10} - 11129932578437 T^{11} + 191889525797724 T^{12} - 1186807376225168 T^{13} + 18425657698557616 T^{14} - 107403884479057789 T^{15} + 1540333692592378371 T^{16} - 8479530196520743306 T^{17} +$$$$11\!\cdots\!33$$$$T^{18} -$$$$59\!\cdots\!86$$$$T^{19} +$$$$77\!\cdots\!10$$$$T^{20} -$$$$38\!\cdots\!39$$$$T^{21} +$$$$47\!\cdots\!63$$$$T^{22} -$$$$22\!\cdots\!65$$$$T^{23} +$$$$27\!\cdots\!73$$$$T^{24} -$$$$12\!\cdots\!57$$$$T^{25} +$$$$35\!\cdots\!17$$$$p T^{26} -$$$$63\!\cdots\!60$$$$T^{27} +$$$$73\!\cdots\!48$$$$T^{28} -$$$$30\!\cdots\!31$$$$T^{29} +$$$$34\!\cdots\!40$$$$T^{30} -$$$$14\!\cdots\!09$$$$T^{31} +$$$$15\!\cdots\!34$$$$T^{32} -$$$$60\!\cdots\!63$$$$T^{33} +$$$$65\!\cdots\!42$$$$T^{34} -$$$$60\!\cdots\!63$$$$p T^{35} +$$$$15\!\cdots\!34$$$$p^{2} T^{36} -$$$$14\!\cdots\!09$$$$p^{3} T^{37} +$$$$34\!\cdots\!40$$$$p^{4} T^{38} -$$$$30\!\cdots\!31$$$$p^{5} T^{39} +$$$$73\!\cdots\!48$$$$p^{6} T^{40} -$$$$63\!\cdots\!60$$$$p^{7} T^{41} +$$$$35\!\cdots\!17$$$$p^{9} T^{42} -$$$$12\!\cdots\!57$$$$p^{9} T^{43} +$$$$27\!\cdots\!73$$$$p^{10} T^{44} -$$$$22\!\cdots\!65$$$$p^{11} T^{45} +$$$$47\!\cdots\!63$$$$p^{12} T^{46} -$$$$38\!\cdots\!39$$$$p^{13} T^{47} +$$$$77\!\cdots\!10$$$$p^{14} T^{48} -$$$$59\!\cdots\!86$$$$p^{15} T^{49} +$$$$11\!\cdots\!33$$$$p^{16} T^{50} - 8479530196520743306 p^{17} T^{51} + 1540333692592378371 p^{18} T^{52} - 107403884479057789 p^{19} T^{53} + 18425657698557616 p^{20} T^{54} - 1186807376225168 p^{21} T^{55} + 191889525797724 p^{22} T^{56} - 11129932578437 p^{23} T^{57} + 1693715300257 p^{24} T^{58} - 85586432053 p^{25} T^{59} + 12247835435 p^{26} T^{60} - 515441351 p^{27} T^{61} + 69323008 p^{28} T^{62} - 2270638 p^{29} T^{63} + 286828 p^{30} T^{64} - 6487 p^{31} T^{65} + 769 p^{32} T^{66} - 9 p^{33} T^{67} + p^{34} T^{68}$$
43 $$1 + 24 T + 979 T^{2} + 19034 T^{3} + 452040 T^{4} + 7447837 T^{5} + 133244039 T^{6} + 1918589863 T^{7} + 28478110841 T^{8} + 366428978081 T^{9} + 4739605850898 T^{10} + 55417389238569 T^{11} + 643191107259901 T^{12} + 6923816271846543 T^{13} + 73521205349565280 T^{14} + 736329217147607518 T^{15} + 7253336371324966375 T^{16} + 68168404310261258132 T^{17} +$$$$62\!\cdots\!80$$$$T^{18} +$$$$55\!\cdots\!96$$$$T^{19} +$$$$48\!\cdots\!00$$$$T^{20} +$$$$41\!\cdots\!87$$$$T^{21} +$$$$79\!\cdots\!92$$$$p T^{22} +$$$$27\!\cdots\!60$$$$T^{23} +$$$$21\!\cdots\!23$$$$T^{24} +$$$$16\!\cdots\!50$$$$T^{25} +$$$$12\!\cdots\!57$$$$T^{26} +$$$$96\!\cdots\!31$$$$T^{27} +$$$$70\!\cdots\!65$$$$T^{28} +$$$$50\!\cdots\!45$$$$T^{29} +$$$$36\!\cdots\!37$$$$T^{30} +$$$$25\!\cdots\!95$$$$T^{31} +$$$$17\!\cdots\!42$$$$T^{32} +$$$$11\!\cdots\!91$$$$T^{33} +$$$$76\!\cdots\!76$$$$T^{34} +$$$$11\!\cdots\!91$$$$p T^{35} +$$$$17\!\cdots\!42$$$$p^{2} T^{36} +$$$$25\!\cdots\!95$$$$p^{3} T^{37} +$$$$36\!\cdots\!37$$$$p^{4} T^{38} +$$$$50\!\cdots\!45$$$$p^{5} T^{39} +$$$$70\!\cdots\!65$$$$p^{6} T^{40} +$$$$96\!\cdots\!31$$$$p^{7} T^{41} +$$$$12\!\cdots\!57$$$$p^{8} T^{42} +$$$$16\!\cdots\!50$$$$p^{9} T^{43} +$$$$21\!\cdots\!23$$$$p^{10} T^{44} +$$$$27\!\cdots\!60$$$$p^{11} T^{45} +$$$$79\!\cdots\!92$$$$p^{13} T^{46} +$$$$41\!\cdots\!87$$$$p^{13} T^{47} +$$$$48\!\cdots\!00$$$$p^{14} T^{48} +$$$$55\!\cdots\!96$$$$p^{15} T^{49} +$$$$62\!\cdots\!80$$$$p^{16} T^{50} + 68168404310261258132 p^{17} T^{51} + 7253336371324966375 p^{18} T^{52} + 736329217147607518 p^{19} T^{53} + 73521205349565280 p^{20} T^{54} + 6923816271846543 p^{21} T^{55} + 643191107259901 p^{22} T^{56} + 55417389238569 p^{23} T^{57} + 4739605850898 p^{24} T^{58} + 366428978081 p^{25} T^{59} + 28478110841 p^{26} T^{60} + 1918589863 p^{27} T^{61} + 133244039 p^{28} T^{62} + 7447837 p^{29} T^{63} + 452040 p^{30} T^{64} + 19034 p^{31} T^{65} + 979 p^{32} T^{66} + 24 p^{33} T^{67} + p^{34} T^{68}$$
47 $$1 + 29 T + 1505 T^{2} + 34065 T^{3} + 1024904 T^{4} + 19308800 T^{5} + 433690343 T^{6} + 7066312513 T^{7} + 130337094916 T^{8} + 1883600016016 T^{9} + 29965007651097 T^{10} + 391002935656678 T^{11} + 5525947767424554 T^{12} + 65971005034005850 T^{13} + 844772893613484580 T^{14} + 9321349883395368615 T^{15} +$$$$10\!\cdots\!13$$$$T^{16} +$$$$11\!\cdots\!71$$$$T^{17} +$$$$12\!\cdots\!46$$$$T^{18} +$$$$11\!\cdots\!38$$$$T^{19} +$$$$12\!\cdots\!32$$$$T^{20} +$$$$11\!\cdots\!28$$$$T^{21} +$$$$10\!\cdots\!39$$$$T^{22} +$$$$91\!\cdots\!71$$$$T^{23} +$$$$83\!\cdots\!50$$$$T^{24} +$$$$68\!\cdots\!30$$$$T^{25} +$$$$59\!\cdots\!12$$$$T^{26} +$$$$46\!\cdots\!93$$$$T^{27} +$$$$37\!\cdots\!75$$$$T^{28} +$$$$28\!\cdots\!30$$$$T^{29} +$$$$22\!\cdots\!98$$$$T^{30} +$$$$15\!\cdots\!11$$$$T^{31} +$$$$11\!\cdots\!19$$$$T^{32} +$$$$81\!\cdots\!74$$$$T^{33} +$$$$58\!\cdots\!00$$$$T^{34} +$$$$81\!\cdots\!74$$$$p T^{35} +$$$$11\!\cdots\!19$$$$p^{2} T^{36} +$$$$15\!\cdots\!11$$$$p^{3} T^{37} +$$$$22\!\cdots\!98$$$$p^{4} T^{38} +$$$$28\!\cdots\!30$$$$p^{5} T^{39} +$$$$37\!\cdots\!75$$$$p^{6} T^{40} +$$$$46\!\cdots\!93$$$$p^{7} T^{41} +$$$$59\!\cdots\!12$$$$p^{8} T^{42} +$$$$68\!\cdots\!30$$$$p^{9} T^{43} +$$$$83\!\cdots\!50$$$$p^{10} T^{44} +$$$$91\!\cdots\!71$$$$p^{11} T^{45} +$$$$10\!\cdots\!39$$$$p^{12} T^{46} +$$$$11\!\cdots\!28$$$$p^{13} T^{47} +$$$$12\!\cdots\!32$$$$p^{14} T^{48} +$$$$11\!\cdots\!38$$$$p^{15} T^{49} +$$$$12\!\cdots\!46$$$$p^{16} T^{50} +$$$$11\!\cdots\!71$$$$p^{17} T^{51} +$$$$10\!\cdots\!13$$$$p^{18} T^{52} + 9321349883395368615 p^{19} T^{53} + 844772893613484580 p^{20} T^{54} + 65971005034005850 p^{21} T^{55} + 5525947767424554 p^{22} T^{56} + 391002935656678 p^{23} T^{57} + 29965007651097 p^{24} T^{58} + 1883600016016 p^{25} T^{59} + 130337094916 p^{26} T^{60} + 7066312513 p^{27} T^{61} + 433690343 p^{28} T^{62} + 19308800 p^{29} T^{63} + 1024904 p^{30} T^{64} + 34065 p^{31} T^{65} + 1505 p^{32} T^{66} + 29 p^{33} T^{67} + p^{34} T^{68}$$
53 $$1 + 9 T + 922 T^{2} + 7676 T^{3} + 418667 T^{4} + 3249644 T^{5} + 125653511 T^{6} + 916151638 T^{7} + 28207813113 T^{8} + 194529146314 T^{9} + 5076384313733 T^{10} + 33310784258378 T^{11} + 765414007752700 T^{12} + 4802330305520228 T^{13} + 99640893513361682 T^{14} + 600007282092741047 T^{15} + 11439967143758068804 T^{16} + 66304809220927157286 T^{17} +$$$$11\!\cdots\!22$$$$T^{18} +$$$$65\!\cdots\!24$$$$T^{19} +$$$$10\!\cdots\!81$$$$T^{20} +$$$$59\!\cdots\!49$$$$T^{21} +$$$$93\!\cdots\!92$$$$T^{22} +$$$$48\!\cdots\!90$$$$T^{23} +$$$$73\!\cdots\!02$$$$T^{24} +$$$$37\!\cdots\!58$$$$T^{25} +$$$$53\!\cdots\!47$$$$T^{26} +$$$$26\!\cdots\!37$$$$T^{27} +$$$$35\!\cdots\!33$$$$T^{28} +$$$$17\!\cdots\!13$$$$T^{29} +$$$$22\!\cdots\!71$$$$T^{30} +$$$$10\!\cdots\!19$$$$T^{31} +$$$$13\!\cdots\!83$$$$T^{32} +$$$$58\!\cdots\!62$$$$T^{33} +$$$$72\!\cdots\!04$$$$T^{34} +$$$$58\!\cdots\!62$$$$p T^{35} +$$$$13\!\cdots\!83$$$$p^{2} T^{36} +$$$$10\!\cdots\!19$$$$p^{3} T^{37} +$$$$22\!\cdots\!71$$$$p^{4} T^{38} +$$$$17\!\cdots\!13$$$$p^{5} T^{39} +$$$$35\!\cdots\!33$$$$p^{6} T^{40} +$$$$26\!\cdots\!37$$$$p^{7} T^{41} +$$$$53\!\cdots\!47$$$$p^{8} T^{42} +$$$$37\!\cdots\!58$$$$p^{9} T^{43} +$$$$73\!\cdots\!02$$$$p^{10} T^{44} +$$$$48\!\cdots\!90$$$$p^{11} T^{45} +$$$$93\!\cdots\!92$$$$p^{12} T^{46} +$$$$59\!\cdots\!49$$$$p^{13} T^{47} +$$$$10\!\cdots\!81$$$$p^{14} T^{48} +$$$$65\!\cdots\!24$$$$p^{15} T^{49} +$$$$11\!\cdots\!22$$$$p^{16} T^{50} + 66304809220927157286 p^{17} T^{51} + 11439967143758068804 p^{18} T^{52} + 600007282092741047 p^{19} T^{53} + 99640893513361682 p^{20} T^{54} + 4802330305520228 p^{21} T^{55} + 765414007752700 p^{22} T^{56} + 33310784258378 p^{23} T^{57} + 5076384313733 p^{24} T^{58} + 194529146314 p^{25} T^{59} + 28207813113 p^{26} T^{60} + 916151638 p^{27} T^{61} + 125653511 p^{28} T^{62} + 3249644 p^{29} T^{63} + 418667 p^{30} T^{64} + 7676 p^{31} T^{65} + 922 p^{32} T^{66} + 9 p^{33} T^{67} + p^{34} T^{68}$$
59 $$1 + 28 T + 1453 T^{2} + 33928 T^{3} + 1013810 T^{4} + 20348799 T^{5} + 455670885 T^{6} + 8048939274 T^{7} + 149117815324 T^{8} + 2361108508622 T^{9} + 38038567175388 T^{10} + 547805577401153 T^{11} + 7902622601242655 T^{12} + 104722227502270962 T^{13} + 1378784010555603434 T^{14} + 16971235437278349042 T^{15} +$$$$20\!\cdots\!90$$$$T^{16} +$$$$23\!\cdots\!20$$$$T^{17} +$$$$27\!\cdots\!46$$$$T^{18} +$$$$29\!\cdots\!91$$$$T^{19} +$$$$31\!\cdots\!95$$$$T^{20} +$$$$32\!\cdots\!74$$$$T^{21} +$$$$32\!\cdots\!92$$$$T^{22} +$$$$32\!\cdots\!14$$$$T^{23} +$$$$30\!\cdots\!91$$$$T^{24} +$$$$28\!\cdots\!25$$$$T^{25} +$$$$26\!\cdots\!08$$$$T^{26} +$$$$23\!\cdots\!24$$$$T^{27} +$$$$20\!\cdots\!35$$$$T^{28} +$$$$17\!\cdots\!86$$$$T^{29} +$$$$14\!\cdots\!14$$$$T^{30} +$$$$12\!\cdots\!74$$$$T^{31} +$$$$99\!\cdots\!32$$$$T^{32} +$$$$78\!\cdots\!82$$$$T^{33} +$$$$61\!\cdots\!14$$$$T^{34} +$$$$78\!\cdots\!82$$$$p T^{35} +$$$$99\!\cdots\!32$$$$p^{2} T^{36} +$$$$12\!\cdots\!74$$$$p^{3} T^{37} +$$$$14\!\cdots\!14$$$$p^{4} T^{38} +$$$$17\!\cdots\!86$$$$p^{5} T^{39} +$$$$20\!\cdots\!35$$$$p^{6} T^{40} +$$$$23\!\cdots\!24$$$$p^{7} T^{41} +$$$$26\!\cdots\!08$$$$p^{8} T^{42} +$$$$28\!\cdots\!25$$$$p^{9} T^{43} +$$$$30\!\cdots\!91$$$$p^{10} T^{44} +$$$$32\!\cdots\!14$$$$p^{11} T^{45} +$$$$32\!\cdots\!92$$$$p^{12} T^{46} +$$$$32\!\cdots\!74$$$$p^{13} T^{47} +$$$$31\!\cdots\!95$$$$p^{14} T^{48} +$$$$29\!\cdots\!91$$$$p^{15} T^{49} +$$$$27\!\cdots\!46$$$$p^{16} T^{50} +$$$$23\!\cdots\!20$$$$p^{17} T^{51} +$$$$20\!\cdots\!90$$$$p^{18} T^{52} + 16971235437278349042 p^{19} T^{53} + 1378784010555603434 p^{20} T^{54} + 104722227502270962 p^{21} T^{55} + 7902622601242655 p^{22} T^{56} + 547805577401153 p^{23} T^{57} + 38038567175388 p^{24} T^{58} + 2361108508622 p^{25} T^{59} + 149117815324 p^{26} T^{60} + 8048939274 p^{27} T^{61} + 455670885 p^{28} T^{62} + 20348799 p^{29} T^{63} + 1013810 p^{30} T^{64} + 33928 p^{31} T^{65} + 1453 p^{32} T^{66} + 28 p^{33} T^{67} + p^{34} T^{68}$$
61 $$1 - 23 T + 1303 T^{2} - 24894 T^{3} + 788202 T^{4} - 12939048 T^{5} + 298620377 T^{6} - 4308504677 T^{7} + 80275284061 T^{8} - 1034273309302 T^{9} + 16404991077520 T^{10} - 190928151444802 T^{11} + 2662506637271407 T^{12} - 28227716796547820 T^{13} + 353664484481835840 T^{14} - 3436322710883452580 T^{15} + 39296289747316971248 T^{16} -$$$$35\!\cdots\!38$$$$T^{17} + 60860900538909147429 p T^{18} -$$$$30\!\cdots\!57$$$$T^{19} +$$$$30\!\cdots\!84$$$$T^{20} -$$$$22\!\cdots\!77$$$$T^{21} +$$$$21\!\cdots\!02$$$$T^{22} -$$$$14\!\cdots\!90$$$$T^{23} +$$$$13\!\cdots\!17$$$$T^{24} -$$$$85\!\cdots\!13$$$$T^{25} +$$$$72\!\cdots\!78$$$$T^{26} -$$$$42\!\cdots\!21$$$$T^{27} +$$$$36\!\cdots\!97$$$$T^{28} -$$$$19\!\cdots\!82$$$$T^{29} +$$$$16\!\cdots\!91$$$$T^{30} -$$$$84\!\cdots\!90$$$$T^{31} +$$$$79\!\cdots\!03$$$$T^{32} -$$$$39\!\cdots\!30$$$$T^{33} +$$$$43\!\cdots\!32$$$$T^{34} -$$$$39\!\cdots\!30$$$$p T^{35} +$$$$79\!\cdots\!03$$$$p^{2} T^{36} -$$$$84\!\cdots\!90$$$$p^{3} T^{37} +$$$$16\!\cdots\!91$$$$p^{4} T^{38} -$$$$19\!\cdots\!82$$$$p^{5} T^{39} +$$$$36\!\cdots\!97$$$$p^{6} T^{40} -$$$$42\!\cdots\!21$$$$p^{7} T^{41} +$$$$72\!\cdots\!78$$$$p^{8} T^{42} -$$$$85\!\cdots\!13$$$$p^{9} T^{43} +$$$$13\!\cdots\!17$$$$p^{10} T^{44} -$$$$14\!\cdots\!90$$$$p^{11} T^{45} +$$$$21\!\cdots\!02$$$$p^{12} T^{46} -$$$$22\!\cdots\!77$$$$p^{13} T^{47} +$$$$30\!\cdots\!84$$$$p^{14} T^{48} -$$$$30\!\cdots\!57$$$$p^{15} T^{49} + 60860900538909147429 p^{17} T^{50} -$$$$35\!\cdots\!38$$$$p^{17} T^{51} + 39296289747316971248 p^{18} T^{52} - 3436322710883452580 p^{19} T^{53} + 353664484481835840 p^{20} T^{54} - 28227716796547820 p^{21} T^{55} + 2662506637271407 p^{22} T^{56} - 190928151444802 p^{23} T^{57} + 16404991077520 p^{24} T^{58} - 1034273309302 p^{25} T^{59} + 80275284061 p^{26} T^{60} - 4308504677 p^{27} T^{61} + 298620377 p^{28} T^{62} - 12939048 p^{29} T^{63} + 788202 p^{30} T^{64} - 24894 p^{31} T^{65} + 1303 p^{32} T^{66} - 23 p^{33} T^{67} + p^{34} T^{68}$$
67 $$1 + 51 T + 2630 T^{2} + 88459 T^{3} + 2816713 T^{4} + 73333771 T^{5} + 1801149519 T^{6} + 39054787873 T^{7} + 803020732935 T^{8} + 15112913976303 T^{9} + 271327346355858 T^{10} + 4549983288160958 T^{11} + 73173374475841937 T^{12} + 1113233479731414885 T^{13} + 16314399928407937388 T^{14} +$$$$22\!\cdots\!04$$$$T^{15} +$$$$30\!\cdots\!68$$$$T^{16} +$$$$40\!\cdots\!39$$$$T^{17} +$$$$50\!\cdots\!69$$$$T^{18} +$$$$61\!\cdots\!37$$$$T^{19} +$$$$72\!\cdots\!88$$$$T^{20} +$$$$82\!\cdots\!33$$$$T^{21} +$$$$91\!\cdots\!66$$$$T^{22} +$$$$98\!\cdots\!70$$$$T^{23} +$$$$10\!\cdots\!10$$$$T^{24} +$$$$10\!\cdots\!66$$$$T^{25} +$$$$10\!\cdots\!51$$$$T^{26} +$$$$10\!\cdots\!01$$$$T^{27} +$$$$98\!\cdots\!97$$$$T^{28} +$$$$91\!\cdots\!95$$$$T^{29} +$$$$82\!\cdots\!57$$$$T^{30} +$$$$73\!\cdots\!93$$$$T^{31} +$$$$63\!\cdots\!71$$$$T^{32} +$$$$53\!\cdots\!44$$$$T^{33} +$$$$44\!\cdots\!28$$$$T^{34} +$$$$53\!\cdots\!44$$$$p T^{35} +$$$$63\!\cdots\!71$$$$p^{2} T^{36} +$$$$73\!\cdots\!93$$$$p^{3} T^{37} +$$$$82\!\cdots\!57$$$$p^{4} T^{38} +$$$$91\!\cdots\!95$$$$p^{5} T^{39} +$$$$98\!\cdots\!97$$$$p^{6} T^{40} +$$$$10\!\cdots\!01$$$$p^{7} T^{41} +$$$$10\!\cdots\!51$$$$p^{8} T^{42} +$$$$10\!\cdots\!66$$$$p^{9} T^{43} +$$$$10\!\cdots\!10$$$$p^{10} T^{44} +$$$$98\!\cdots\!70$$$$p^{11} T^{45} +$$$$91\!\cdots\!66$$$$p^{12} T^{46} +$$$$82\!\cdots\!33$$$$p^{13} T^{47} +$$$$72\!\cdots\!88$$$$p^{14} T^{48} +$$$$61\!\cdots\!37$$$$p^{15} T^{49} +$$$$50\!\cdots\!69$$$$p^{16} T^{50} +$$$$40\!\cdots\!39$$$$p^{17} T^{51} +$$$$30\!\cdots\!68$$$$p^{18} T^{52} +$$$$22\!\cdots\!04$$$$p^{19} T^{53} + 16314399928407937388 p^{20} T^{54} + 1113233479731414885 p^{21} T^{55} + 73173374475841937 p^{22} T^{56} + 4549983288160958 p^{23} T^{57} + 271327346355858 p^{24} T^{58} + 15112913976303 p^{25} T^{59} + 803020732935 p^{26} T^{60} + 39054787873 p^{27} T^{61} + 1801149519 p^{28} T^{62} + 73333771 p^{29} T^{63} + 2816713 p^{30} T^{64} + 88459 p^{31} T^{65} + 2630 p^{32} T^{66} + 51 p^{33} T^{67} + p^{34} T^{68}$$
71 $$1 + 31 T + 1784 T^{2} + 47313 T^{3} + 1570535 T^{4} + 36213379 T^{5} + 906414519 T^{6} + 18486739455 T^{7} + 385433247954 T^{8} + 7062285845964 T^{9} + 128794730858302 T^{10} + 2147986602744018 T^{11} + 35230330923732949 T^{12} + 540508421038223190 T^{13} + 8113041836307331117 T^{14} +$$$$11\!\cdots\!90$$$$T^{15} +$$$$16\!\cdots\!83$$$$T^{16} +$$$$21\!\cdots\!50$$$$T^{17} +$$$$27\!\cdots\!13$$$$T^{18} +$$$$34\!\cdots\!94$$$$T^{19} +$$$$42\!\cdots\!86$$$$T^{20} +$$$$49\!\cdots\!86$$$$T^{21} +$$$$57\!\cdots\!89$$$$T^{22} +$$$$63\!\cdots\!26$$$$T^{23} +$$$$69\!\cdots\!35$$$$T^{24} +$$$$73\!\cdots\!10$$$$T^{25} +$$$$75\!\cdots\!12$$$$T^{26} +$$$$76\!\cdots\!76$$$$T^{27} +$$$$75\!\cdots\!26$$$$T^{28} +$$$$72\!\cdots\!09$$$$T^{29} +$$$$67\!\cdots\!00$$$$T^{30} +$$$$61\!\cdots\!60$$$$T^{31} +$$$$55\!\cdots\!33$$$$T^{32} +$$$$48\!\cdots\!15$$$$T^{33} +$$$$41\!\cdots\!07$$$$T^{34} +$$$$48\!\cdots\!15$$$$p T^{35} +$$$$55\!\cdots\!33$$$$p^{2} T^{36} +$$$$61\!\cdots\!60$$$$p^{3} T^{37} +$$$$67\!\cdots\!00$$$$p^{4} T^{38} +$$$$72\!\cdots\!09$$$$p^{5} T^{39} +$$$$75\!\cdots\!26$$$$p^{6} T^{40} +$$$$76\!\cdots\!76$$$$p^{7} T^{41} +$$$$75\!\cdots\!12$$$$p^{8} T^{42} +$$$$73\!\cdots\!10$$$$p^{9} T^{43} +$$$$69\!\cdots\!35$$$$p^{10} T^{44} +$$$$63\!\cdots\!26$$$$p^{11} T^{45} +$$$$57\!\cdots\!89$$$$p^{12} T^{46} +$$$$49\!\cdots\!86$$$$p^{13} T^{47} +$$$$42\!\cdots\!86$$$$p^{14} T^{48} +$$$$34\!\cdots\!94$$$$p^{15} T^{49} +$$$$27\!\cdots\!13$$$$p^{16} T^{50} +$$$$21\!\cdots\!50$$$$p^{17} T^{51} +$$$$16\!\cdots\!83$$$$p^{18} T^{52} +$$$$11\!\cdots\!90$$$$p^{19} T^{53} + 8113041836307331117 p^{20} T^{54} + 540508421038223190 p^{21} T^{55} + 35230330923732949 p^{22} T^{56} + 2147986602744018 p^{23} T^{57} + 128794730858302 p^{24} T^{58} + 7062285845964 p^{25} T^{59} + 385433247954 p^{26} T^{60} + 18486739455 p^{27} T^{61} + 906414519 p^{28} T^{62} + 36213379 p^{29} T^{63} + 1570535 p^{30} T^{64} + 47313 p^{31} T^{65} + 1784 p^{32} T^{66} + 31 p^{33} T^{67} + p^{34} T^{68}$$
73 $$1 + 26 T + 1609 T^{2} + 35449 T^{3} + 1254039 T^{4} + 24223858 T^{5} + 637484075 T^{6} + 11047437688 T^{7} + 239109319208 T^{8} + 3778799209500 T^{9} + 70820523809135 T^{10} + 1033119875303243 T^{11} + 17288529851577745 T^{12} + 234969521331288742 T^{13} + 3582156902217044039 T^{14} + 45688837177621085195 T^{15} +$$$$64\!\cdots\!29$$$$T^{16} +$$$$77\!\cdots\!20$$$$T^{17} +$$$$10\!\cdots\!42$$$$T^{18} +$$$$11\!\cdots\!07$$$$T^{19} +$$$$14\!\cdots\!31$$$$T^{20} +$$$$15\!\cdots\!20$$$$T^{21} +$$$$18\!\cdots\!87$$$$T^{22} +$$$$18\!\cdots\!18$$$$T^{23} +$$$$21\!\cdots\!15$$$$T^{24} +$$$$20\!\cdots\!10$$$$T^{25} +$$$$22\!\cdots\!12$$$$T^{26} +$$$$21\!\cdots\!40$$$$T^{27} +$$$$21\!\cdots\!47$$$$T^{28} +$$$$19\!\cdots\!55$$$$T^{29} +$$$$19\!\cdots\!55$$$$T^{30} +$$$$16\!\cdots\!12$$$$T^{31} +$$$$15\!\cdots\!43$$$$T^{32} +$$$$13\!\cdots\!02$$$$T^{33} +$$$$11\!\cdots\!19$$$$T^{34} +$$$$13\!\cdots\!02$$$$p T^{35} +$$$$15\!\cdots\!43$$$$p^{2} T^{36} +$$$$16\!\cdots\!12$$$$p^{3} T^{37} +$$$$19\!\cdots\!55$$$$p^{4} T^{38} +$$$$19\!\cdots\!55$$$$p^{5} T^{39} +$$$$21\!\cdots\!47$$$$p^{6} T^{40} +$$$$21\!\cdots\!40$$$$p^{7} T^{41} +$$$$22\!\cdots\!12$$$$p^{8} T^{42} +$$$$20\!\cdots\!10$$$$p^{9} T^{43} +$$$$21\!\cdots\!15$$$$p^{10} T^{44} +$$$$18\!\cdots\!18$$$$p^{11} T^{45} +$$$$18\!\cdots\!87$$$$p^{12} T^{46} +$$$$15\!\cdots\!20$$$$p^{13} T^{47} +$$$$14\!\cdots\!31$$$$p^{14} T^{48} +$$$$11\!\cdots\!07$$$$p^{15} T^{49} +$$$$10\!\cdots\!42$$$$p^{16} T^{50} +$$$$77\!\cdots\!20$$$$p^{17} T^{51} +$$$$64\!\cdots\!29$$$$p^{18} T^{52} + 45688837177621085195 p^{19} T^{53} + 3582156902217044039 p^{20} T^{54} + 234969521331288742 p^{21} T^{55} + 17288529851577745 p^{22} T^{56} + 1033119875303243 p^{23} T^{57} + 70820523809135 p^{24} T^{58} + 3778799209500 p^{25} T^{59} + 239109319208 p^{26} T^{60} + 11047437688 p^{27} T^{61} + 637484075 p^{28} T^{62} + 24223858 p^{29} T^{63} + 1254039 p^{30} T^{64} + 35449 p^{31} T^{65} + 1609 p^{32} T^{66} + 26 p^{33} T^{67} + p^{34} T^{68}$$
79 $$1 + 36 T + 2156 T^{2} + 60476 T^{3} + 2112974 T^{4} + 49325269 T^{5} + 1292662321 T^{6} + 26115982295 T^{7} + 564148353412 T^{8} + 10122290428967 T^{9} + 189175869112473 T^{10} + 3070273824944881 T^{11} + 51118305774830016 T^{12} + 9630831117122463 p T^{13} + 11509212286023851205 T^{14} +$$$$15\!\cdots\!71$$$$T^{15} +$$$$22\!\cdots\!88$$$$T^{16} +$$$$28\!\cdots\!49$$$$T^{17} +$$$$37\!\cdots\!06$$$$T^{18} +$$$$45\!\cdots\!59$$$$T^{19} +$$$$55\!\cdots\!17$$$$T^{20} +$$$$63\!\cdots\!21$$$$T^{21} +$$$$92\!\cdots\!45$$$$p T^{22} +$$$$80\!\cdots\!64$$$$T^{23} +$$$$88\!\cdots\!78$$$$T^{24} +$$$$93\!\cdots\!63$$$$T^{25} +$$$$98\!\cdots\!64$$$$T^{26} +$$$$99\!\cdots\!67$$$$T^{27} +$$$$10\!\cdots\!17$$$$T^{28} +$$$$97\!\cdots\!53$$$$T^{29} +$$$$94\!\cdots\!05$$$$T^{30} +$$$$88\!\cdots\!24$$$$T^{31} +$$$$83\!\cdots\!72$$$$T^{32} +$$$$75\!\cdots\!36$$$$T^{33} +$$$$68\!\cdots\!53$$$$T^{34} +$$$$75\!\cdots\!36$$$$p T^{35} +$$$$83\!\cdots\!72$$$$p^{2} T^{36} +$$$$88\!\cdots\!24$$$$p^{3} T^{37} +$$$$94\!\cdots\!05$$$$p^{4} T^{38} +$$$$97\!\cdots\!53$$$$p^{5} T^{39} +$$$$10\!\cdots\!17$$$$p^{6} T^{40} +$$$$99\!\cdots\!67$$$$p^{7} T^{41} +$$$$98\!\cdots\!64$$$$p^{8} T^{42} +$$$$93\!\cdots\!63$$$$p^{9} T^{43} +$$$$88\!\cdots\!78$$$$p^{10} T^{44} +$$$$80\!\cdots\!64$$$$p^{11} T^{45} +$$$$92\!\cdots\!45$$$$p^{13} T^{46} +$$$$63\!\cdots\!21$$$$p^{13} T^{47} +$$$$55\!\cdots\!17$$$$p^{14} T^{48} +$$$$45\!\cdots\!59$$$$p^{15} T^{49} +$$$$37\!\cdots\!06$$$$p^{16} T^{50} +$$$$28\!\cdots\!49$$$$p^{17} T^{51} +$$$$22\!\cdots\!88$$$$p^{18} T^{52} +$$$$15\!\cdots\!71$$$$p^{19} T^{53} + 11509212286023851205 p^{20} T^{54} + 9630831117122463 p^{22} T^{55} + 51118305774830016 p^{22} T^{56} + 3070273824944881 p^{23} T^{57} + 189175869112473 p^{24} T^{58} + 10122290428967 p^{25} T^{59} + 564148353412 p^{26} T^{60} + 26115982295 p^{27} T^{61} + 1292662321 p^{28} T^{62} + 49325269 p^{29} T^{63} + 2112974 p^{30} T^{64} + 60476 p^{31} T^{65} + 2156 p^{32} T^{66} + 36 p^{33} T^{67} + p^{34} T^{68}$$
83 $$1 + 10 T + 1812 T^{2} + 16076 T^{3} + 1604816 T^{4} + 12577112 T^{5} + 928218891 T^{6} + 6382896170 T^{7} + 395320777312 T^{8} + 2363500173876 T^{9} + 132545027501172 T^{10} + 681052540458185 T^{11} + 36529620832403922 T^{12} + 159077294593168921 T^{13} + 8531051224318480931 T^{14} + 30978585050332212336 T^{15} +$$$$17\!\cdots\!63$$$$T^{16} +$$$$51\!\cdots\!28$$$$T^{17} +$$$$30\!\cdots\!91$$$$T^{18} +$$$$73\!\cdots\!83$$$$T^{19} +$$$$49\!\cdots\!36$$$$T^{20} +$$$$92\!\cdots\!30$$$$T^{21} +$$$$70\!\cdots\!15$$$$T^{22} +$$$$10\!\cdots\!83$$$$T^{23} +$$$$92\!\cdots\!75$$$$T^{24} +$$$$10\!\cdots\!06$$$$T^{25} +$$$$11\!\cdots\!78$$$$T^{26} +$$$$89\!\cdots\!34$$$$T^{27} +$$$$12\!\cdots\!49$$$$T^{28} +$$$$74\!\cdots\!95$$$$T^{29} +$$$$12\!\cdots\!83$$$$T^{30} +$$$$58\!\cdots\!77$$$$T^{31} +$$$$11\!\cdots\!38$$$$T^{32} +$$$$46\!\cdots\!22$$$$T^{33} +$$$$99\!\cdots\!62$$$$T^{34} +$$$$46\!\cdots\!22$$$$p T^{35} +$$$$11\!\cdots\!38$$$$p^{2} T^{36} +$$$$58\!\cdots\!77$$$$p^{3} T^{37} +$$$$12\!\cdots\!83$$$$p^{4} T^{38} +$$$$74\!\cdots\!95$$$$p^{5} T^{39} +$$$$12\!\cdots\!49$$$$p^{6} T^{40} +$$$$89\!\cdots\!34$$$$p^{7} T^{41} +$$$$11\!\cdots\!78$$$$p^{8} T^{42} +$$$$10\!\cdots\!06$$$$p^{9} T^{43} +$$$$92\!\cdots\!75$$$$p^{10} T^{44} +$$$$10\!\cdots\!83$$$$p^{11} T^{45} +$$$$70\!\cdots\!15$$$$p^{12} T^{46} +$$$$92\!\cdots\!30$$$$p^{13} T^{47} +$$$$49\!\cdots\!36$$$$p^{14} T^{48} +$$$$73\!\cdots\!83$$$$p^{15} T^{49} +$$$$30\!\cdots\!91$$$$p^{16} T^{50} +$$$$51\!\cdots\!28$$$$p^{17} T^{51} +$$$$17\!\cdots\!63$$$$p^{18} T^{52} + 30978585050332212336 p^{19} T^{53} + 8531051224318480931 p^{20} T^{54} + 159077294593168921 p^{21} T^{55} + 36529620832403922 p^{22} T^{56} + 681052540458185 p^{23} T^{57} + 132545027501172 p^{24} T^{58} + 2363500173876 p^{25} T^{59} + 395320777312 p^{26} T^{60} + 6382896170 p^{27} T^{61} + 928218891 p^{28} T^{62} + 12577112 p^{29} T^{63} + 1604816 p^{30} T^{64} + 16076 p^{31} T^{65} + 1812 p^{32} T^{66} + 10 p^{33} T^{67} + p^{34} T^{68}$$
89 $$1 + 34 T + 1868 T^{2} + 50617 T^{3} + 1664028 T^{4} + 38171604 T^{5} + 961463451 T^{6} + 19365085104 T^{7} + 409347345630 T^{8} + 7414964994475 T^{9} + 137700964182447 T^{10} + 2281283264681373 T^{11} + 38237446457281635 T^{12} + 586555371020589291 T^{13} + 9031502463268996478 T^{14} +$$$$12\!\cdots\!87$$$$T^{15} +$$$$18\!\cdots\!83$$$$T^{16} +$$$$25\!\cdots\!61$$$$T^{17} +$$$$33\!\cdots\!68$$$$T^{18} +$$$$43\!\cdots\!34$$$$T^{19} +$$$$54\!\cdots\!58$$$$T^{20} +$$$$66\!\cdots\!84$$$$T^{21} +$$$$80\!\cdots\!79$$$$T^{22} +$$$$93\!\cdots\!64$$$$T^{23} +$$$$10\!\cdots\!06$$$$T^{24} +$$$$12\!\cdots\!96$$$$T^{25} +$$$$13\!\cdots\!92$$$$T^{26} +$$$$14\!\cdots\!86$$$$T^{27} +$$$$15\!\cdots\!84$$$$T^{28} +$$$$15\!\cdots\!74$$$$T^{29} +$$$$16\!\cdots\!79$$$$T^{30} +$$$$16\!\cdots\!01$$$$T^{31} +$$$$15\!\cdots\!20$$$$T^{32} +$$$$15\!\cdots\!89$$$$T^{33} +$$$$14\!\cdots\!94$$$$T^{34} +$$$$15\!\cdots\!89$$$$p T^{35} +$$$$15\!\cdots\!20$$$$p^{2} T^{36} +$$$$16\!\cdots\!01$$$$p^{3} T^{37} +$$$$16\!\cdots\!79$$$$p^{4} T^{38} +$$$$15\!\cdots\!74$$$$p^{5} T^{39} +$$$$15\!\cdots\!84$$$$p^{6} T^{40} +$$$$14\!\cdots\!86$$$$p^{7} T^{41} +$$$$13\!\cdots\!92$$$$p^{8} T^{42} +$$$$12\!\cdots\!96$$$$p^{9} T^{43} +$$$$10\!\cdots\!06$$$$p^{10} T^{44} +$$$$93\!\cdots\!64$$$$p^{11} T^{45} +$$$$80\!\cdots\!79$$$$p^{12} T^{46} +$$$$66\!\cdots\!84$$$$p^{13} T^{47} +$$$$54\!\cdots\!58$$$$p^{14} T^{48} +$$$$43\!\cdots\!34$$$$p^{15} T^{49} +$$$$33\!\cdots\!68$$$$p^{16} T^{50} +$$$$25\!\cdots\!61$$$$p^{17} T^{51} +$$$$18\!\cdots\!83$$$$p^{18} T^{52} +$$$$12\!\cdots\!87$$$$p^{19} T^{53} + 9031502463268996478 p^{20} T^{54} + 586555371020589291 p^{21} T^{55} + 38237446457281635 p^{22} T^{56} + 2281283264681373 p^{23} T^{57} + 137700964182447 p^{24} T^{58} + 7414964994475 p^{25} T^{59} + 409347345630 p^{26} T^{60} + 19365085104 p^{27} T^{61} + 961463451 p^{28} T^{62} + 38171604 p^{29} T^{63} + 1664028 p^{30} T^{64} + 50617 p^{31} T^{65} + 1868 p^{32} T^{66} + 34 p^{33} T^{67} + p^{34} T^{68}$$
97 $$1 + 56 T + 3143 T^{2} + 115813 T^{3} + 4010963 T^{4} + 113957215 T^{5} + 3036817998 T^{6} + 71516814623 T^{7} + 1589476024614 T^{8} + 32315194148940 T^{9} + 624190190156970 T^{10} + 11242884560882531 T^{11} + 193539174727232380 T^{12} + 3144688300619247222 T^{13} + 49086116890414815259 T^{14} +$$$$72\!\cdots\!68$$$$T^{15} +$$$$10\!\cdots\!88$$$$T^{16} +$$$$14\!\cdots\!93$$$$T^{17} +$$$$19\!\cdots\!06$$$$T^{18} +$$$$24\!\cdots\!70$$$$T^{19} +$$$$30\!\cdots\!65$$$$T^{20} +$$$$37\!\cdots\!97$$$$T^{21} +$$$$44\!\cdots\!62$$$$T^{22} +$$$$51\!\cdots\!52$$$$T^{23} +$$$$58\!\cdots\!84$$$$T^{24} +$$$$64\!\cdots\!79$$$$T^{25} +$$$$70\!\cdots\!13$$$$T^{26} +$$$$76\!\cdots\!98$$$$T^{27} +$$$$81\!\cdots\!49$$$$T^{28} +$$$$86\!\cdots\!31$$$$T^{29} +$$$$89\!\cdots\!12$$$$T^{30} +$$$$92\!\cdots\!62$$$$T^{31} +$$$$94\!\cdots\!95$$$$T^{32} +$$$$94\!\cdots\!20$$$$T^{33} +$$$$93\!\cdots\!92$$$$T^{34} +$$$$94\!\cdots\!20$$$$p T^{35} +$$$$94\!\cdots\!95$$$$p^{2} T^{36} +$$$$92\!\cdots\!62$$$$p^{3} T^{37} +$$$$89\!\cdots\!12$$$$p^{4} T^{38} +$$$$86\!\cdots\!31$$$$p^{5} T^{39} +$$$$81\!\cdots\!49$$$$p^{6} T^{40} +$$$$76\!\cdots\!98$$$$p^{7} T^{41} +$$$$70\!\cdots\!13$$$$p^{8} T^{42} +$$$$64\!\cdots\!79$$$$p^{9} T^{43} +$$$$58\!\cdots\!84$$$$p^{10} T^{44} +$$$$51\!\cdots\!52$$$$p^{11} T^{45} +$$$$44\!\cdots\!62$$$$p^{12} T^{46} +$$$$37\!\cdots\!97$$$$p^{13} T^{47} +$$$$30\!\cdots\!65$$$$p^{14} T^{48} +$$$$24\!\cdots\!70$$$$p^{15} T^{49} +$$$$19\!\cdots\!06$$$$p^{16} T^{50} +$$$$14\!\cdots\!93$$$$p^{17} T^{51} +$$$$10\!\cdots\!88$$$$p^{18} T^{52} +$$$$72\!\cdots\!68$$$$p^{19} T^{53} + 49086116890414815259 p^{20} T^{54} + 3144688300619247222 p^{21} T^{55} + 193539174727232380 p^{22} T^{56} + 11242884560882531 p^{23} T^{57} + 624190190156970 p^{24} T^{58} + 32315194148940 p^{25} T^{59} + 1589476024614 p^{26} T^{60} + 71516814623 p^{27} T^{61} + 3036817998 p^{28} T^{62} + 113957215 p^{29} T^{63} + 4010963 p^{30} T^{64} + 115813 p^{31} T^{65} + 3143 p^{32} T^{66} + 56 p^{33} T^{67} + p^{34} T^{68}$$
\begin{aligned} L(s) = \prod_p \ \prod_{j=1}^{68} (1 - \alpha_{j,p}\, p^{-s})^{-1} \end{aligned}