# Properties

 Degree 68 Conductor $13^{34} \cdot 31^{34}$ Sign $1$ Motivic weight 1 Primitive no Self-dual yes Analytic rank 0

# Origins of factors

## Dirichlet series

 L(s)  = 1 + 6·2-s − 2·3-s + 4-s − 5·5-s − 12·6-s − 2·7-s − 58·8-s + 16·9-s − 30·10-s − 5·11-s − 2·12-s − 17·13-s − 12·14-s + 10·15-s − 72·16-s − 8·17-s + 96·18-s + 3·19-s − 5·20-s + 4·21-s − 30·22-s − 14·23-s + 116·24-s + 42·25-s − 102·26-s − 24·27-s − 2·28-s + ⋯
 L(s)  = 1 + 4.24·2-s − 1.15·3-s + 1/2·4-s − 2.23·5-s − 4.89·6-s − 0.755·7-s − 20.5·8-s + 16/3·9-s − 9.48·10-s − 1.50·11-s − 0.577·12-s − 4.71·13-s − 3.20·14-s + 2.58·15-s − 18·16-s − 1.94·17-s + 22.6·18-s + 0.688·19-s − 1.11·20-s + 0.872·21-s − 6.39·22-s − 2.91·23-s + 23.6·24-s + 42/5·25-s − 20.0·26-s − 4.61·27-s − 0.377·28-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut &\left(13^{34} \cdot 31^{34}\right)^{s/2} \, \Gamma_{\C}(s)^{34} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut &\left(13^{34} \cdot 31^{34}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{34} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}

## Invariants

 $$d$$ = $$68$$ $$N$$ = $$13^{34} \cdot 31^{34}$$ $$\varepsilon$$ = $1$ motivic weight = $$1$$ character : induced by $\chi_{403} (1, \cdot )$ primitive : no self-dual : yes analytic rank = $$0$$ Selberg data = $$(68,\ 13^{34} \cdot 31^{34} ,\ ( \ : [1/2]^{34} ),\ 1 )$$ $$L(1)$$ $$\approx$$ $$0.119315$$ $$L(\frac12)$$ $$\approx$$ $$0.119315$$ $$L(\frac{3}{2})$$ not available $$L(1)$$ not available

## Euler product

$L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1}$where, for $p \notin \{13,\;31\}$,$$F_p(T)$$ is a polynomial of degree 68. If $p \in \{13,\;31\}$, then $F_p(T)$ is a polynomial of degree at most 67.
$p$$F_p(T)$
bad13 $$( 1 + T + T^{2} )^{17}$$
31 $$1 + 9 T + 127 T^{2} + 668 T^{3} + 7612 T^{4} + 34364 T^{5} + 447 p^{2} T^{6} + 1976504 T^{7} + 21698358 T^{8} + 86444776 T^{9} + 914802261 T^{10} + 3509985538 T^{11} + 36838354637 T^{12} + 137357411362 T^{13} + 1353309154850 T^{14} + 4676578184964 T^{15} + 44642258355308 T^{16} + 148990922271327 T^{17} + 44642258355308 p T^{18} + 4676578184964 p^{2} T^{19} + 1353309154850 p^{3} T^{20} + 137357411362 p^{4} T^{21} + 36838354637 p^{5} T^{22} + 3509985538 p^{6} T^{23} + 914802261 p^{7} T^{24} + 86444776 p^{8} T^{25} + 21698358 p^{9} T^{26} + 1976504 p^{10} T^{27} + 447 p^{13} T^{28} + 34364 p^{12} T^{29} + 7612 p^{13} T^{30} + 668 p^{14} T^{31} + 127 p^{15} T^{32} + 9 p^{16} T^{33} + p^{17} T^{34}$$
good2 $$( 1 - 3 T + 13 T^{2} - 17 p T^{3} + 3 p^{5} T^{4} - 109 p T^{5} + 249 p T^{6} - 253 p^{2} T^{7} + 505 p^{2} T^{8} - 933 p^{2} T^{9} + 1685 p^{2} T^{10} - 11525 T^{11} + 9607 p T^{12} - 15375 p T^{13} + 11999 p^{2} T^{14} - 72479 T^{15} + 53411 p T^{16} - 152849 T^{17} + 53411 p^{2} T^{18} - 72479 p^{2} T^{19} + 11999 p^{5} T^{20} - 15375 p^{5} T^{21} + 9607 p^{6} T^{22} - 11525 p^{6} T^{23} + 1685 p^{9} T^{24} - 933 p^{10} T^{25} + 505 p^{11} T^{26} - 253 p^{12} T^{27} + 249 p^{12} T^{28} - 109 p^{13} T^{29} + 3 p^{18} T^{30} - 17 p^{15} T^{31} + 13 p^{15} T^{32} - 3 p^{16} T^{33} + p^{17} T^{34} )^{2}$$
3 $$1 + 2 T - 4 p T^{2} - 32 T^{3} + 50 T^{4} + 206 T^{5} - 8 p T^{6} - 221 p T^{7} - 508 T^{8} + 310 p T^{9} + 604 p T^{10} + 445 T^{11} - 1928 T^{12} - 3767 T^{13} - 2800 T^{14} + 6946 T^{15} - 745 T^{16} - 10220 T^{17} + 137072 T^{18} + 29989 p T^{19} - 728540 T^{20} - 683087 T^{21} + 1374097 T^{22} + 1989658 T^{23} + 295091 T^{24} - 1396181 T^{25} - 1386476 T^{26} + 3224902 T^{27} - 6083110 T^{28} - 53819248 T^{29} - 88266293 T^{30} + 66398986 p T^{31} + 819170702 T^{32} - 244990070 T^{33} - 3225381803 T^{34} - 244990070 p T^{35} + 819170702 p^{2} T^{36} + 66398986 p^{4} T^{37} - 88266293 p^{4} T^{38} - 53819248 p^{5} T^{39} - 6083110 p^{6} T^{40} + 3224902 p^{7} T^{41} - 1386476 p^{8} T^{42} - 1396181 p^{9} T^{43} + 295091 p^{10} T^{44} + 1989658 p^{11} T^{45} + 1374097 p^{12} T^{46} - 683087 p^{13} T^{47} - 728540 p^{14} T^{48} + 29989 p^{16} T^{49} + 137072 p^{16} T^{50} - 10220 p^{17} T^{51} - 745 p^{18} T^{52} + 6946 p^{19} T^{53} - 2800 p^{20} T^{54} - 3767 p^{21} T^{55} - 1928 p^{22} T^{56} + 445 p^{23} T^{57} + 604 p^{25} T^{58} + 310 p^{26} T^{59} - 508 p^{26} T^{60} - 221 p^{28} T^{61} - 8 p^{29} T^{62} + 206 p^{29} T^{63} + 50 p^{30} T^{64} - 32 p^{31} T^{65} - 4 p^{33} T^{66} + 2 p^{33} T^{67} + p^{34} T^{68}$$
5 $$1 + p T - 17 T^{2} - 192 T^{3} - 249 T^{4} + 2599 T^{5} + 10718 T^{6} - 6718 T^{7} - 5647 p^{2} T^{8} - 250046 T^{9} + 827838 T^{10} + 158253 p^{2} T^{11} + 878578 T^{12} - 1183111 p^{2} T^{13} - 60590204 T^{14} + 104808083 T^{15} + 596550528 T^{16} + 299274641 T^{17} - 3369341328 T^{18} - 1470273097 p T^{19} + 8968419059 T^{20} + 58879072051 T^{21} + 8323218709 p T^{22} - 274779365374 T^{23} - 682512083397 T^{24} + 419950454942 T^{25} + 4414386719009 T^{26} + 4987602447267 T^{27} - 15030597597237 T^{28} - 10113940838024 p T^{29} - 6524256744303 T^{30} + 245198279465559 T^{31} + 438320853024162 T^{32} - 502122703207842 T^{33} - 3025265679447859 T^{34} - 502122703207842 p T^{35} + 438320853024162 p^{2} T^{36} + 245198279465559 p^{3} T^{37} - 6524256744303 p^{4} T^{38} - 10113940838024 p^{6} T^{39} - 15030597597237 p^{6} T^{40} + 4987602447267 p^{7} T^{41} + 4414386719009 p^{8} T^{42} + 419950454942 p^{9} T^{43} - 682512083397 p^{10} T^{44} - 274779365374 p^{11} T^{45} + 8323218709 p^{13} T^{46} + 58879072051 p^{13} T^{47} + 8968419059 p^{14} T^{48} - 1470273097 p^{16} T^{49} - 3369341328 p^{16} T^{50} + 299274641 p^{17} T^{51} + 596550528 p^{18} T^{52} + 104808083 p^{19} T^{53} - 60590204 p^{20} T^{54} - 1183111 p^{23} T^{55} + 878578 p^{22} T^{56} + 158253 p^{25} T^{57} + 827838 p^{24} T^{58} - 250046 p^{25} T^{59} - 5647 p^{28} T^{60} - 6718 p^{27} T^{61} + 10718 p^{28} T^{62} + 2599 p^{29} T^{63} - 249 p^{30} T^{64} - 192 p^{31} T^{65} - 17 p^{32} T^{66} + p^{34} T^{67} + p^{34} T^{68}$$
7 $$1 + 2 T - 59 T^{2} - 16 p T^{3} + 1712 T^{4} + 3065 T^{5} - 4630 p T^{6} - 55077 T^{7} + 64135 p T^{8} + 735576 T^{9} - 4879942 T^{10} - 7798290 T^{11} + 44443253 T^{12} + 67824520 T^{13} - 368104491 T^{14} - 488447082 T^{15} + 3024117907 T^{16} + 2904883356 T^{17} - 25605220498 T^{18} - 2078940881 p T^{19} + 217235451830 T^{20} + 71035620954 T^{21} - 1775324450846 T^{22} - 459122949991 T^{23} + 13829681223509 T^{24} + 78078573831 p^{2} T^{25} - 103908715311370 T^{26} - 29556242841688 T^{27} + 763173700178564 T^{28} + 180102328506694 T^{29} - 5509617682242148 T^{30} - 805303083215017 T^{31} + 39181665887513956 T^{32} + 1874996264149038 T^{33} - 275504494529437582 T^{34} + 1874996264149038 p T^{35} + 39181665887513956 p^{2} T^{36} - 805303083215017 p^{3} T^{37} - 5509617682242148 p^{4} T^{38} + 180102328506694 p^{5} T^{39} + 763173700178564 p^{6} T^{40} - 29556242841688 p^{7} T^{41} - 103908715311370 p^{8} T^{42} + 78078573831 p^{11} T^{43} + 13829681223509 p^{10} T^{44} - 459122949991 p^{11} T^{45} - 1775324450846 p^{12} T^{46} + 71035620954 p^{13} T^{47} + 217235451830 p^{14} T^{48} - 2078940881 p^{16} T^{49} - 25605220498 p^{16} T^{50} + 2904883356 p^{17} T^{51} + 3024117907 p^{18} T^{52} - 488447082 p^{19} T^{53} - 368104491 p^{20} T^{54} + 67824520 p^{21} T^{55} + 44443253 p^{22} T^{56} - 7798290 p^{23} T^{57} - 4879942 p^{24} T^{58} + 735576 p^{25} T^{59} + 64135 p^{27} T^{60} - 55077 p^{27} T^{61} - 4630 p^{29} T^{62} + 3065 p^{29} T^{63} + 1712 p^{30} T^{64} - 16 p^{32} T^{65} - 59 p^{32} T^{66} + 2 p^{33} T^{67} + p^{34} T^{68}$$
11 $$1 + 5 T - 82 T^{2} - 395 T^{3} + 3406 T^{4} + 14889 T^{5} - 8909 p T^{6} - 362576 T^{7} + 2228187 T^{8} + 6643935 T^{9} - 42457400 T^{10} - 102904669 T^{11} + 696830756 T^{12} + 1470089730 T^{13} - 10122032937 T^{14} - 20244347583 T^{15} + 135466905062 T^{16} + 272437354223 T^{17} - 1745282928176 T^{18} - 3584288227145 T^{19} + 22253806564984 T^{20} + 45094804131632 T^{21} - 282400620829495 T^{22} - 520133062647564 T^{23} + 3537072987707134 T^{24} + 5302734083527318 T^{25} - 43083958246885172 T^{26} - 46976015584973850 T^{27} + 504339329953612233 T^{28} + 358163958452670368 T^{29} - 4271477535979440 p^{3} T^{30} - 2235808202768727410 T^{31} + 62710283176009787380 T^{32} + 7868639664151035432 T^{33} -$$$$68\!\cdots\!80$$$$T^{34} + 7868639664151035432 p T^{35} + 62710283176009787380 p^{2} T^{36} - 2235808202768727410 p^{3} T^{37} - 4271477535979440 p^{7} T^{38} + 358163958452670368 p^{5} T^{39} + 504339329953612233 p^{6} T^{40} - 46976015584973850 p^{7} T^{41} - 43083958246885172 p^{8} T^{42} + 5302734083527318 p^{9} T^{43} + 3537072987707134 p^{10} T^{44} - 520133062647564 p^{11} T^{45} - 282400620829495 p^{12} T^{46} + 45094804131632 p^{13} T^{47} + 22253806564984 p^{14} T^{48} - 3584288227145 p^{15} T^{49} - 1745282928176 p^{16} T^{50} + 272437354223 p^{17} T^{51} + 135466905062 p^{18} T^{52} - 20244347583 p^{19} T^{53} - 10122032937 p^{20} T^{54} + 1470089730 p^{21} T^{55} + 696830756 p^{22} T^{56} - 102904669 p^{23} T^{57} - 42457400 p^{24} T^{58} + 6643935 p^{25} T^{59} + 2228187 p^{26} T^{60} - 362576 p^{27} T^{61} - 8909 p^{29} T^{62} + 14889 p^{29} T^{63} + 3406 p^{30} T^{64} - 395 p^{31} T^{65} - 82 p^{32} T^{66} + 5 p^{33} T^{67} + p^{34} T^{68}$$
17 $$1 + 8 T - 121 T^{2} - 926 T^{3} + 8587 T^{4} + 55924 T^{5} - 441791 T^{6} - 2257756 T^{7} + 17488789 T^{8} + 66744000 T^{9} - 544574110 T^{10} - 1516428597 T^{11} + 13512469358 T^{12} + 27925512926 T^{13} - 270015507881 T^{14} - 461149883148 T^{15} + 4434368482039 T^{16} + 7898314275966 T^{17} - 63629403041992 T^{18} - 143651721777041 T^{19} + 924558927495966 T^{20} + 2289395864626772 T^{21} - 15911459294945837 T^{22} - 1289473317702155 p T^{23} + 309938445017873003 T^{24} - 136416844750455780 T^{25} - 5944168049718882509 T^{26} + 10656799022932128063 T^{27} +$$$$10\!\cdots\!16$$$$T^{28} -$$$$24\!\cdots\!63$$$$T^{29} -$$$$19\!\cdots\!82$$$$T^{30} +$$$$33\!\cdots\!67$$$$T^{31} +$$$$33\!\cdots\!31$$$$T^{32} -$$$$21\!\cdots\!08$$$$T^{33} -$$$$58\!\cdots\!34$$$$T^{34} -$$$$21\!\cdots\!08$$$$p T^{35} +$$$$33\!\cdots\!31$$$$p^{2} T^{36} +$$$$33\!\cdots\!67$$$$p^{3} T^{37} -$$$$19\!\cdots\!82$$$$p^{4} T^{38} -$$$$24\!\cdots\!63$$$$p^{5} T^{39} +$$$$10\!\cdots\!16$$$$p^{6} T^{40} + 10656799022932128063 p^{7} T^{41} - 5944168049718882509 p^{8} T^{42} - 136416844750455780 p^{9} T^{43} + 309938445017873003 p^{10} T^{44} - 1289473317702155 p^{12} T^{45} - 15911459294945837 p^{12} T^{46} + 2289395864626772 p^{13} T^{47} + 924558927495966 p^{14} T^{48} - 143651721777041 p^{15} T^{49} - 63629403041992 p^{16} T^{50} + 7898314275966 p^{17} T^{51} + 4434368482039 p^{18} T^{52} - 461149883148 p^{19} T^{53} - 270015507881 p^{20} T^{54} + 27925512926 p^{21} T^{55} + 13512469358 p^{22} T^{56} - 1516428597 p^{23} T^{57} - 544574110 p^{24} T^{58} + 66744000 p^{25} T^{59} + 17488789 p^{26} T^{60} - 2257756 p^{27} T^{61} - 441791 p^{28} T^{62} + 55924 p^{29} T^{63} + 8587 p^{30} T^{64} - 926 p^{31} T^{65} - 121 p^{32} T^{66} + 8 p^{33} T^{67} + p^{34} T^{68}$$
19 $$1 - 3 T - 130 T^{2} + 25 p T^{3} + 7935 T^{4} - 36350 T^{5} - 820 p^{2} T^{6} + 1818354 T^{7} + 6963209 T^{8} - 66258887 T^{9} - 69061468 T^{10} + 1806123891 T^{11} - 2241428999 T^{12} - 35037524849 T^{13} + 137834107703 T^{14} + 357530169716 T^{15} - 3986907740533 T^{16} + 4604581214954 T^{17} + 71762739892032 T^{18} - 315441918521204 T^{19} - 539994159263388 T^{20} + 414915839967527 p T^{21} - 14482344239253590 T^{22} - 102779565926347237 T^{23} + 674588102277321591 T^{24} - 386600011064788710 T^{25} - 14114455576997898996 T^{26} + 3114646479619703384 p T^{27} +$$$$13\!\cdots\!60$$$$T^{28} -$$$$17\!\cdots\!21$$$$T^{29} +$$$$19\!\cdots\!34$$$$T^{30} +$$$$29\!\cdots\!86$$$$T^{31} -$$$$11\!\cdots\!45$$$$T^{32} -$$$$22\!\cdots\!10$$$$T^{33} +$$$$28\!\cdots\!36$$$$T^{34} -$$$$22\!\cdots\!10$$$$p T^{35} -$$$$11\!\cdots\!45$$$$p^{2} T^{36} +$$$$29\!\cdots\!86$$$$p^{3} T^{37} +$$$$19\!\cdots\!34$$$$p^{4} T^{38} -$$$$17\!\cdots\!21$$$$p^{5} T^{39} +$$$$13\!\cdots\!60$$$$p^{6} T^{40} + 3114646479619703384 p^{8} T^{41} - 14114455576997898996 p^{8} T^{42} - 386600011064788710 p^{9} T^{43} + 674588102277321591 p^{10} T^{44} - 102779565926347237 p^{11} T^{45} - 14482344239253590 p^{12} T^{46} + 414915839967527 p^{14} T^{47} - 539994159263388 p^{14} T^{48} - 315441918521204 p^{15} T^{49} + 71762739892032 p^{16} T^{50} + 4604581214954 p^{17} T^{51} - 3986907740533 p^{18} T^{52} + 357530169716 p^{19} T^{53} + 137834107703 p^{20} T^{54} - 35037524849 p^{21} T^{55} - 2241428999 p^{22} T^{56} + 1806123891 p^{23} T^{57} - 69061468 p^{24} T^{58} - 66258887 p^{25} T^{59} + 6963209 p^{26} T^{60} + 1818354 p^{27} T^{61} - 820 p^{30} T^{62} - 36350 p^{29} T^{63} + 7935 p^{30} T^{64} + 25 p^{32} T^{65} - 130 p^{32} T^{66} - 3 p^{33} T^{67} + p^{34} T^{68}$$
23 $$( 1 + 7 T + 243 T^{2} + 1696 T^{3} + 29291 T^{4} + 198133 T^{5} + 2336492 T^{6} + 15012259 T^{7} + 138472217 T^{8} + 834911697 T^{9} + 6479208573 T^{10} + 36429446720 T^{11} + 248001854907 T^{12} + 1296153663137 T^{13} + 7931754971941 T^{14} + 38445388478024 T^{15} + 214534923619213 T^{16} + 41799379275230 p T^{17} + 214534923619213 p T^{18} + 38445388478024 p^{2} T^{19} + 7931754971941 p^{3} T^{20} + 1296153663137 p^{4} T^{21} + 248001854907 p^{5} T^{22} + 36429446720 p^{6} T^{23} + 6479208573 p^{7} T^{24} + 834911697 p^{8} T^{25} + 138472217 p^{9} T^{26} + 15012259 p^{10} T^{27} + 2336492 p^{11} T^{28} + 198133 p^{12} T^{29} + 29291 p^{13} T^{30} + 1696 p^{14} T^{31} + 243 p^{15} T^{32} + 7 p^{16} T^{33} + p^{17} T^{34} )^{2}$$
29 $$( 1 + 9 T + 298 T^{2} + 2261 T^{3} + 42534 T^{4} + 283693 T^{5} + 3960612 T^{6} + 23872677 T^{7} + 274135283 T^{8} + 1520416236 T^{9} + 15133475004 T^{10} + 78036625941 T^{11} + 23926040828 p T^{12} + 3341298185582 T^{13} + 27035210427221 T^{14} + 121634731733602 T^{15} + 906318284356883 T^{16} + 3801313727255541 T^{17} + 906318284356883 p T^{18} + 121634731733602 p^{2} T^{19} + 27035210427221 p^{3} T^{20} + 3341298185582 p^{4} T^{21} + 23926040828 p^{6} T^{22} + 78036625941 p^{6} T^{23} + 15133475004 p^{7} T^{24} + 1520416236 p^{8} T^{25} + 274135283 p^{9} T^{26} + 23872677 p^{10} T^{27} + 3960612 p^{11} T^{28} + 283693 p^{12} T^{29} + 42534 p^{13} T^{30} + 2261 p^{14} T^{31} + 298 p^{15} T^{32} + 9 p^{16} T^{33} + p^{17} T^{34} )^{2}$$
37 $$1 + 6 T - 274 T^{2} - 2060 T^{3} + 35671 T^{4} + 332317 T^{5} - 2861550 T^{6} - 33814836 T^{7} + 151190350 T^{8} + 2437004583 T^{9} - 4825754747 T^{10} - 130601000475 T^{11} + 17962326284 T^{12} + 5146760883140 T^{13} + 8580055640568 T^{14} - 126108354959295 T^{15} - 603309766007968 T^{16} - 588136279868588 T^{17} + 22558869658626000 T^{18} + 264374533437254192 T^{19} - 292712657705771852 T^{20} - 16700726687318942890 T^{21} - 23485427344364242456 T^{22} +$$$$66\!\cdots\!73$$$$T^{23} +$$$$19\!\cdots\!43$$$$T^{24} -$$$$17\!\cdots\!52$$$$T^{25} -$$$$81\!\cdots\!50$$$$T^{26} +$$$$23\!\cdots\!27$$$$T^{27} +$$$$16\!\cdots\!31$$$$T^{28} +$$$$52\!\cdots\!45$$$$T^{29} +$$$$31\!\cdots\!71$$$$T^{30} -$$$$37\!\cdots\!52$$$$T^{31} -$$$$44\!\cdots\!24$$$$T^{32} +$$$$65\!\cdots\!47$$$$T^{33} +$$$$21\!\cdots\!53$$$$T^{34} +$$$$65\!\cdots\!47$$$$p T^{35} -$$$$44\!\cdots\!24$$$$p^{2} T^{36} -$$$$37\!\cdots\!52$$$$p^{3} T^{37} +$$$$31\!\cdots\!71$$$$p^{4} T^{38} +$$$$52\!\cdots\!45$$$$p^{5} T^{39} +$$$$16\!\cdots\!31$$$$p^{6} T^{40} +$$$$23\!\cdots\!27$$$$p^{7} T^{41} -$$$$81\!\cdots\!50$$$$p^{8} T^{42} -$$$$17\!\cdots\!52$$$$p^{9} T^{43} +$$$$19\!\cdots\!43$$$$p^{10} T^{44} +$$$$66\!\cdots\!73$$$$p^{11} T^{45} - 23485427344364242456 p^{12} T^{46} - 16700726687318942890 p^{13} T^{47} - 292712657705771852 p^{14} T^{48} + 264374533437254192 p^{15} T^{49} + 22558869658626000 p^{16} T^{50} - 588136279868588 p^{17} T^{51} - 603309766007968 p^{18} T^{52} - 126108354959295 p^{19} T^{53} + 8580055640568 p^{20} T^{54} + 5146760883140 p^{21} T^{55} + 17962326284 p^{22} T^{56} - 130601000475 p^{23} T^{57} - 4825754747 p^{24} T^{58} + 2437004583 p^{25} T^{59} + 151190350 p^{26} T^{60} - 33814836 p^{27} T^{61} - 2861550 p^{28} T^{62} + 332317 p^{29} T^{63} + 35671 p^{30} T^{64} - 2060 p^{31} T^{65} - 274 p^{32} T^{66} + 6 p^{33} T^{67} + p^{34} T^{68}$$
41 $$1 + 5 T - 269 T^{2} - 1890 T^{3} + 33337 T^{4} + 306740 T^{5} - 2390588 T^{6} - 29064318 T^{7} + 103809160 T^{8} + 1760048331 T^{9} - 2897122763 T^{10} - 66984482497 T^{11} + 125990557990 T^{12} + 1263275944435 T^{13} - 13451833703613 T^{14} + 12394077904854 T^{15} + 1106299485855098 T^{16} - 1216799652788130 T^{17} - 60751559557191072 T^{18} + 2320517961765935 T^{19} + 2407131021482180156 T^{20} + 2524305333467557103 T^{21} - 74032416785552957279 T^{22} -$$$$17\!\cdots\!63$$$$T^{23} +$$$$18\!\cdots\!23$$$$T^{24} +$$$$77\!\cdots\!26$$$$T^{25} -$$$$46\!\cdots\!93$$$$T^{26} -$$$$19\!\cdots\!50$$$$T^{27} +$$$$22\!\cdots\!29$$$$T^{28} -$$$$22\!\cdots\!88$$$$T^{29} -$$$$19\!\cdots\!85$$$$T^{30} +$$$$38\!\cdots\!77$$$$T^{31} +$$$$13\!\cdots\!66$$$$T^{32} -$$$$88\!\cdots\!54$$$$T^{33} -$$$$66\!\cdots\!24$$$$T^{34} -$$$$88\!\cdots\!54$$$$p T^{35} +$$$$13\!\cdots\!66$$$$p^{2} T^{36} +$$$$38\!\cdots\!77$$$$p^{3} T^{37} -$$$$19\!\cdots\!85$$$$p^{4} T^{38} -$$$$22\!\cdots\!88$$$$p^{5} T^{39} +$$$$22\!\cdots\!29$$$$p^{6} T^{40} -$$$$19\!\cdots\!50$$$$p^{7} T^{41} -$$$$46\!\cdots\!93$$$$p^{8} T^{42} +$$$$77\!\cdots\!26$$$$p^{9} T^{43} +$$$$18\!\cdots\!23$$$$p^{10} T^{44} -$$$$17\!\cdots\!63$$$$p^{11} T^{45} - 74032416785552957279 p^{12} T^{46} + 2524305333467557103 p^{13} T^{47} + 2407131021482180156 p^{14} T^{48} + 2320517961765935 p^{15} T^{49} - 60751559557191072 p^{16} T^{50} - 1216799652788130 p^{17} T^{51} + 1106299485855098 p^{18} T^{52} + 12394077904854 p^{19} T^{53} - 13451833703613 p^{20} T^{54} + 1263275944435 p^{21} T^{55} + 125990557990 p^{22} T^{56} - 66984482497 p^{23} T^{57} - 2897122763 p^{24} T^{58} + 1760048331 p^{25} T^{59} + 103809160 p^{26} T^{60} - 29064318 p^{27} T^{61} - 2390588 p^{28} T^{62} + 306740 p^{29} T^{63} + 33337 p^{30} T^{64} - 1890 p^{31} T^{65} - 269 p^{32} T^{66} + 5 p^{33} T^{67} + p^{34} T^{68}$$
43 $$1 + T - 320 T^{2} + 769 T^{3} + 50415 T^{4} - 284602 T^{5} - 4589706 T^{6} + 44952240 T^{7} + 214244979 T^{8} - 4225743057 T^{9} + 2350888332 T^{10} + 251109506094 T^{11} - 1165070841351 T^{12} - 8239520389603 T^{13} + 91141512914478 T^{14} - 22364647737193 T^{15} - 3696703348635678 T^{16} + 17550868309613796 T^{17} + 50498962938172653 T^{18} - 837063539932592133 T^{19} + 2916529718553311715 T^{20} + 7449753322469717304 T^{21} -$$$$14\!\cdots\!93$$$$T^{22} +$$$$10\!\cdots\!64$$$$T^{23} -$$$$28\!\cdots\!30$$$$T^{24} -$$$$45\!\cdots\!68$$$$T^{25} +$$$$56\!\cdots\!71$$$$T^{26} -$$$$10\!\cdots\!75$$$$T^{27} -$$$$25\!\cdots\!29$$$$T^{28} +$$$$19\!\cdots\!47$$$$T^{29} +$$$$18\!\cdots\!64$$$$T^{30} -$$$$99\!\cdots\!13$$$$T^{31} +$$$$44\!\cdots\!94$$$$T^{32} +$$$$18\!\cdots\!11$$$$T^{33} -$$$$29\!\cdots\!66$$$$T^{34} +$$$$18\!\cdots\!11$$$$p T^{35} +$$$$44\!\cdots\!94$$$$p^{2} T^{36} -$$$$99\!\cdots\!13$$$$p^{3} T^{37} +$$$$18\!\cdots\!64$$$$p^{4} T^{38} +$$$$19\!\cdots\!47$$$$p^{5} T^{39} -$$$$25\!\cdots\!29$$$$p^{6} T^{40} -$$$$10\!\cdots\!75$$$$p^{7} T^{41} +$$$$56\!\cdots\!71$$$$p^{8} T^{42} -$$$$45\!\cdots\!68$$$$p^{9} T^{43} -$$$$28\!\cdots\!30$$$$p^{10} T^{44} +$$$$10\!\cdots\!64$$$$p^{11} T^{45} -$$$$14\!\cdots\!93$$$$p^{12} T^{46} + 7449753322469717304 p^{13} T^{47} + 2916529718553311715 p^{14} T^{48} - 837063539932592133 p^{15} T^{49} + 50498962938172653 p^{16} T^{50} + 17550868309613796 p^{17} T^{51} - 3696703348635678 p^{18} T^{52} - 22364647737193 p^{19} T^{53} + 91141512914478 p^{20} T^{54} - 8239520389603 p^{21} T^{55} - 1165070841351 p^{22} T^{56} + 251109506094 p^{23} T^{57} + 2350888332 p^{24} T^{58} - 4225743057 p^{25} T^{59} + 214244979 p^{26} T^{60} + 44952240 p^{27} T^{61} - 4589706 p^{28} T^{62} - 284602 p^{29} T^{63} + 50415 p^{30} T^{64} + 769 p^{31} T^{65} - 320 p^{32} T^{66} + p^{33} T^{67} + p^{34} T^{68}$$
47 $$( 1 - 8 T + 382 T^{2} - 2708 T^{3} + 72785 T^{4} - 506628 T^{5} + 9504268 T^{6} - 67992488 T^{7} + 956443661 T^{8} - 7098318850 T^{9} + 78718204960 T^{10} - 600946315079 T^{11} + 5498377936985 T^{12} - 900868294788 p T^{13} + 333268164691535 T^{14} - 2521765104244530 T^{15} + 17738098320219901 T^{16} - 128117290634842662 T^{17} + 17738098320219901 p T^{18} - 2521765104244530 p^{2} T^{19} + 333268164691535 p^{3} T^{20} - 900868294788 p^{5} T^{21} + 5498377936985 p^{5} T^{22} - 600946315079 p^{6} T^{23} + 78718204960 p^{7} T^{24} - 7098318850 p^{8} T^{25} + 956443661 p^{9} T^{26} - 67992488 p^{10} T^{27} + 9504268 p^{11} T^{28} - 506628 p^{12} T^{29} + 72785 p^{13} T^{30} - 2708 p^{14} T^{31} + 382 p^{15} T^{32} - 8 p^{16} T^{33} + p^{17} T^{34} )^{2}$$
53 $$1 - 30 T + 120 T^{2} + 4426 T^{3} - 33363 T^{4} - 413707 T^{5} + 3758455 T^{6} + 29495156 T^{7} - 262243861 T^{8} - 1874625319 T^{9} + 13382478451 T^{10} + 98870147565 T^{11} - 450831157116 T^{12} - 4342023346044 T^{13} + 5499626815122 T^{14} + 175191449376368 T^{15} + 173828603550507 T^{16} - 5584696888317629 T^{17} - 7382525415856012 T^{18} + 125587271493176260 T^{19} - 515086467393103615 T^{20} + 3088062255346207301 T^{21} + 57313468003518223560 T^{22} -$$$$70\!\cdots\!42$$$$T^{23} -$$$$34\!\cdots\!30$$$$T^{24} +$$$$71\!\cdots\!80$$$$T^{25} +$$$$70\!\cdots\!14$$$$T^{26} -$$$$44\!\cdots\!92$$$$T^{27} +$$$$18\!\cdots\!76$$$$T^{28} +$$$$19\!\cdots\!78$$$$T^{29} -$$$$15\!\cdots\!95$$$$T^{30} -$$$$72\!\cdots\!86$$$$T^{31} +$$$$18\!\cdots\!53$$$$T^{32} +$$$$13\!\cdots\!15$$$$T^{33} -$$$$11\!\cdots\!62$$$$T^{34} +$$$$13\!\cdots\!15$$$$p T^{35} +$$$$18\!\cdots\!53$$$$p^{2} T^{36} -$$$$72\!\cdots\!86$$$$p^{3} T^{37} -$$$$15\!\cdots\!95$$$$p^{4} T^{38} +$$$$19\!\cdots\!78$$$$p^{5} T^{39} +$$$$18\!\cdots\!76$$$$p^{6} T^{40} -$$$$44\!\cdots\!92$$$$p^{7} T^{41} +$$$$70\!\cdots\!14$$$$p^{8} T^{42} +$$$$71\!\cdots\!80$$$$p^{9} T^{43} -$$$$34\!\cdots\!30$$$$p^{10} T^{44} -$$$$70\!\cdots\!42$$$$p^{11} T^{45} + 57313468003518223560 p^{12} T^{46} + 3088062255346207301 p^{13} T^{47} - 515086467393103615 p^{14} T^{48} + 125587271493176260 p^{15} T^{49} - 7382525415856012 p^{16} T^{50} - 5584696888317629 p^{17} T^{51} + 173828603550507 p^{18} T^{52} + 175191449376368 p^{19} T^{53} + 5499626815122 p^{20} T^{54} - 4342023346044 p^{21} T^{55} - 450831157116 p^{22} T^{56} + 98870147565 p^{23} T^{57} + 13382478451 p^{24} T^{58} - 1874625319 p^{25} T^{59} - 262243861 p^{26} T^{60} + 29495156 p^{27} T^{61} + 3758455 p^{28} T^{62} - 413707 p^{29} T^{63} - 33363 p^{30} T^{64} + 4426 p^{31} T^{65} + 120 p^{32} T^{66} - 30 p^{33} T^{67} + p^{34} T^{68}$$
59 $$1 + 9 T - 546 T^{2} - 101 p T^{3} + 145058 T^{4} + 1946177 T^{5} - 24594243 T^{6} - 421744118 T^{7} + 2873112697 T^{8} + 68390281109 T^{9} - 218275535837 T^{10} - 8833804337191 T^{11} + 4971501620263 T^{12} + 941750096454585 T^{13} + 1519314970488496 T^{14} - 84532250113496944 T^{15} - 309487197712887010 T^{16} + 6441450069789727151 T^{17} + 37840816794311224701 T^{18} -$$$$41\!\cdots\!89$$$$T^{19} -$$$$36\!\cdots\!31$$$$T^{20} +$$$$22\!\cdots\!53$$$$T^{21} +$$$$28\!\cdots\!01$$$$T^{22} -$$$$88\!\cdots\!01$$$$T^{23} -$$$$19\!\cdots\!36$$$$T^{24} +$$$$19\!\cdots\!16$$$$T^{25} +$$$$11\!\cdots\!98$$$$T^{26} +$$$$69\!\cdots\!40$$$$T^{27} -$$$$64\!\cdots\!54$$$$T^{28} -$$$$10\!\cdots\!31$$$$T^{29} +$$$$32\!\cdots\!77$$$$T^{30} +$$$$59\!\cdots\!89$$$$T^{31} -$$$$26\!\cdots\!66$$$$p T^{32} -$$$$14\!\cdots\!26$$$$T^{33} +$$$$85\!\cdots\!39$$$$T^{34} -$$$$14\!\cdots\!26$$$$p T^{35} -$$$$26\!\cdots\!66$$$$p^{3} T^{36} +$$$$59\!\cdots\!89$$$$p^{3} T^{37} +$$$$32\!\cdots\!77$$$$p^{4} T^{38} -$$$$10\!\cdots\!31$$$$p^{5} T^{39} -$$$$64\!\cdots\!54$$$$p^{6} T^{40} +$$$$69\!\cdots\!40$$$$p^{7} T^{41} +$$$$11\!\cdots\!98$$$$p^{8} T^{42} +$$$$19\!\cdots\!16$$$$p^{9} T^{43} -$$$$19\!\cdots\!36$$$$p^{10} T^{44} -$$$$88\!\cdots\!01$$$$p^{11} T^{45} +$$$$28\!\cdots\!01$$$$p^{12} T^{46} +$$$$22\!\cdots\!53$$$$p^{13} T^{47} -$$$$36\!\cdots\!31$$$$p^{14} T^{48} -$$$$41\!\cdots\!89$$$$p^{15} T^{49} + 37840816794311224701 p^{16} T^{50} + 6441450069789727151 p^{17} T^{51} - 309487197712887010 p^{18} T^{52} - 84532250113496944 p^{19} T^{53} + 1519314970488496 p^{20} T^{54} + 941750096454585 p^{21} T^{55} + 4971501620263 p^{22} T^{56} - 8833804337191 p^{23} T^{57} - 218275535837 p^{24} T^{58} + 68390281109 p^{25} T^{59} + 2873112697 p^{26} T^{60} - 421744118 p^{27} T^{61} - 24594243 p^{28} T^{62} + 1946177 p^{29} T^{63} + 145058 p^{30} T^{64} - 101 p^{32} T^{65} - 546 p^{32} T^{66} + 9 p^{33} T^{67} + p^{34} T^{68}$$
61 $$( 1 + 14 T + 516 T^{2} + 5030 T^{3} + 119655 T^{4} + 904147 T^{5} + 18621269 T^{6} + 116841744 T^{7} + 2291710057 T^{8} + 12339276672 T^{9} + 235709699663 T^{10} + 1101753737173 T^{11} + 340373422894 p T^{12} + 85775094902733 T^{13} + 1602926502646145 T^{14} + 6015166223424689 T^{15} + 109883325564335544 T^{16} + 384434889250424780 T^{17} + 109883325564335544 p T^{18} + 6015166223424689 p^{2} T^{19} + 1602926502646145 p^{3} T^{20} + 85775094902733 p^{4} T^{21} + 340373422894 p^{6} T^{22} + 1101753737173 p^{6} T^{23} + 235709699663 p^{7} T^{24} + 12339276672 p^{8} T^{25} + 2291710057 p^{9} T^{26} + 116841744 p^{10} T^{27} + 18621269 p^{11} T^{28} + 904147 p^{12} T^{29} + 119655 p^{13} T^{30} + 5030 p^{14} T^{31} + 516 p^{15} T^{32} + 14 p^{16} T^{33} + p^{17} T^{34} )^{2}$$
67 $$1 + 31 T - 260 T^{2} - 14573 T^{3} + 75381 T^{4} + 4610815 T^{5} - 19876971 T^{6} - 1048667701 T^{7} + 5355393531 T^{8} + 188650789838 T^{9} - 1226439325265 T^{10} - 27124653999206 T^{11} + 234441143923550 T^{12} + 3122624928700109 T^{13} - 37072012055866384 T^{14} - 273751237507883041 T^{15} + 4901314081818881490 T^{16} + 14880275239705859185 T^{17} -$$$$54\!\cdots\!73$$$$T^{18} +$$$$29\!\cdots\!46$$$$T^{19} +$$$$50\!\cdots\!04$$$$T^{20} -$$$$20\!\cdots\!89$$$$T^{21} -$$$$37\!\cdots\!13$$$$T^{22} +$$$$31\!\cdots\!41$$$$T^{23} +$$$$20\!\cdots\!86$$$$T^{24} -$$$$33\!\cdots\!59$$$$T^{25} -$$$$39\!\cdots\!25$$$$T^{26} +$$$$28\!\cdots\!78$$$$T^{27} -$$$$73\!\cdots\!66$$$$T^{28} -$$$$19\!\cdots\!48$$$$T^{29} +$$$$12\!\cdots\!11$$$$T^{30} +$$$$95\!\cdots\!24$$$$T^{31} -$$$$12\!\cdots\!82$$$$T^{32} -$$$$23\!\cdots\!70$$$$T^{33} +$$$$92\!\cdots\!06$$$$T^{34} -$$$$23\!\cdots\!70$$$$p T^{35} -$$$$12\!\cdots\!82$$$$p^{2} T^{36} +$$$$95\!\cdots\!24$$$$p^{3} T^{37} +$$$$12\!\cdots\!11$$$$p^{4} T^{38} -$$$$19\!\cdots\!48$$$$p^{5} T^{39} -$$$$73\!\cdots\!66$$$$p^{6} T^{40} +$$$$28\!\cdots\!78$$$$p^{7} T^{41} -$$$$39\!\cdots\!25$$$$p^{8} T^{42} -$$$$33\!\cdots\!59$$$$p^{9} T^{43} +$$$$20\!\cdots\!86$$$$p^{10} T^{44} +$$$$31\!\cdots\!41$$$$p^{11} T^{45} -$$$$37\!\cdots\!13$$$$p^{12} T^{46} -$$$$20\!\cdots\!89$$$$p^{13} T^{47} +$$$$50\!\cdots\!04$$$$p^{14} T^{48} +$$$$29\!\cdots\!46$$$$p^{15} T^{49} -$$$$54\!\cdots\!73$$$$p^{16} T^{50} + 14880275239705859185 p^{17} T^{51} + 4901314081818881490 p^{18} T^{52} - 273751237507883041 p^{19} T^{53} - 37072012055866384 p^{20} T^{54} + 3122624928700109 p^{21} T^{55} + 234441143923550 p^{22} T^{56} - 27124653999206 p^{23} T^{57} - 1226439325265 p^{24} T^{58} + 188650789838 p^{25} T^{59} + 5355393531 p^{26} T^{60} - 1048667701 p^{27} T^{61} - 19876971 p^{28} T^{62} + 4610815 p^{29} T^{63} + 75381 p^{30} T^{64} - 14573 p^{31} T^{65} - 260 p^{32} T^{66} + 31 p^{33} T^{67} + p^{34} T^{68}$$
71 $$1 - T - 667 T^{2} + 1624 T^{3} + 218074 T^{4} - 793589 T^{5} - 46505127 T^{6} + 208869102 T^{7} + 7344016035 T^{8} - 36125081523 T^{9} - 941528528030 T^{10} + 4623987699222 T^{11} + 105962993643897 T^{12} - 491382995600814 T^{13} - 10969341415123958 T^{14} + 48794655560005550 T^{15} + 1048401512056028707 T^{16} - 4771253338416333749 T^{17} - 91646821705879838348 T^{18} +$$$$44\!\cdots\!24$$$$T^{19} +$$$$74\!\cdots\!76$$$$T^{20} -$$$$37\!\cdots\!57$$$$T^{21} -$$$$58\!\cdots\!26$$$$T^{22} +$$$$28\!\cdots\!26$$$$T^{23} +$$$$45\!\cdots\!22$$$$T^{24} -$$$$21\!\cdots\!30$$$$T^{25} -$$$$33\!\cdots\!26$$$$T^{26} +$$$$14\!\cdots\!58$$$$T^{27} +$$$$23\!\cdots\!08$$$$T^{28} -$$$$81\!\cdots\!31$$$$T^{29} -$$$$16\!\cdots\!72$$$$T^{30} +$$$$35\!\cdots\!59$$$$T^{31} +$$$$11\!\cdots\!33$$$$T^{32} -$$$$80\!\cdots\!29$$$$T^{33} -$$$$84\!\cdots\!98$$$$T^{34} -$$$$80\!\cdots\!29$$$$p T^{35} +$$$$11\!\cdots\!33$$$$p^{2} T^{36} +$$$$35\!\cdots\!59$$$$p^{3} T^{37} -$$$$16\!\cdots\!72$$$$p^{4} T^{38} -$$$$81\!\cdots\!31$$$$p^{5} T^{39} +$$$$23\!\cdots\!08$$$$p^{6} T^{40} +$$$$14\!\cdots\!58$$$$p^{7} T^{41} -$$$$33\!\cdots\!26$$$$p^{8} T^{42} -$$$$21\!\cdots\!30$$$$p^{9} T^{43} +$$$$45\!\cdots\!22$$$$p^{10} T^{44} +$$$$28\!\cdots\!26$$$$p^{11} T^{45} -$$$$58\!\cdots\!26$$$$p^{12} T^{46} -$$$$37\!\cdots\!57$$$$p^{13} T^{47} +$$$$74\!\cdots\!76$$$$p^{14} T^{48} +$$$$44\!\cdots\!24$$$$p^{15} T^{49} - 91646821705879838348 p^{16} T^{50} - 4771253338416333749 p^{17} T^{51} + 1048401512056028707 p^{18} T^{52} + 48794655560005550 p^{19} T^{53} - 10969341415123958 p^{20} T^{54} - 491382995600814 p^{21} T^{55} + 105962993643897 p^{22} T^{56} + 4623987699222 p^{23} T^{57} - 941528528030 p^{24} T^{58} - 36125081523 p^{25} T^{59} + 7344016035 p^{26} T^{60} + 208869102 p^{27} T^{61} - 46505127 p^{28} T^{62} - 793589 p^{29} T^{63} + 218074 p^{30} T^{64} + 1624 p^{31} T^{65} - 667 p^{32} T^{66} - p^{33} T^{67} + p^{34} T^{68}$$
73 $$1 + 10 T - 410 T^{2} - 8346 T^{3} + 37601 T^{4} + 2393163 T^{5} + 15708825 T^{6} - 281370382 T^{7} - 5240793321 T^{8} - 9550115277 T^{9} + 612114580649 T^{10} + 97702845214 p T^{11} - 2213845525696 T^{12} - 810621522438433 T^{13} - 7687054843147426 T^{14} + 6006882332023427 T^{15} + 795304334996814432 T^{16} + 7216451973884020444 T^{17} + 138739076152682693 T^{18} -$$$$61\!\cdots\!48$$$$T^{19} -$$$$59\!\cdots\!99$$$$T^{20} -$$$$95\!\cdots\!40$$$$T^{21} +$$$$36\!\cdots\!61$$$$T^{22} +$$$$42\!\cdots\!72$$$$T^{23} +$$$$15\!\cdots\!50$$$$T^{24} -$$$$15\!\cdots\!16$$$$T^{25} -$$$$24\!\cdots\!49$$$$T^{26} -$$$$13\!\cdots\!68$$$$T^{27} +$$$$19\!\cdots\!15$$$$T^{28} +$$$$10\!\cdots\!56$$$$T^{29} +$$$$86\!\cdots\!27$$$$T^{30} +$$$$31\!\cdots\!61$$$$T^{31} -$$$$12\!\cdots\!43$$$$T^{32} -$$$$30\!\cdots\!53$$$$T^{33} -$$$$30\!\cdots\!04$$$$T^{34} -$$$$30\!\cdots\!53$$$$p T^{35} -$$$$12\!\cdots\!43$$$$p^{2} T^{36} +$$$$31\!\cdots\!61$$$$p^{3} T^{37} +$$$$86\!\cdots\!27$$$$p^{4} T^{38} +$$$$10\!\cdots\!56$$$$p^{5} T^{39} +$$$$19\!\cdots\!15$$$$p^{6} T^{40} -$$$$13\!\cdots\!68$$$$p^{7} T^{41} -$$$$24\!\cdots\!49$$$$p^{8} T^{42} -$$$$15\!\cdots\!16$$$$p^{9} T^{43} +$$$$15\!\cdots\!50$$$$p^{10} T^{44} +$$$$42\!\cdots\!72$$$$p^{11} T^{45} +$$$$36\!\cdots\!61$$$$p^{12} T^{46} -$$$$95\!\cdots\!40$$$$p^{13} T^{47} -$$$$59\!\cdots\!99$$$$p^{14} T^{48} -$$$$61\!\cdots\!48$$$$p^{15} T^{49} + 138739076152682693 p^{16} T^{50} + 7216451973884020444 p^{17} T^{51} + 795304334996814432 p^{18} T^{52} + 6006882332023427 p^{19} T^{53} - 7687054843147426 p^{20} T^{54} - 810621522438433 p^{21} T^{55} - 2213845525696 p^{22} T^{56} + 97702845214 p^{24} T^{57} + 612114580649 p^{24} T^{58} - 9550115277 p^{25} T^{59} - 5240793321 p^{26} T^{60} - 281370382 p^{27} T^{61} + 15708825 p^{28} T^{62} + 2393163 p^{29} T^{63} + 37601 p^{30} T^{64} - 8346 p^{31} T^{65} - 410 p^{32} T^{66} + 10 p^{33} T^{67} + p^{34} T^{68}$$
79 $$1 + 23 T - 235 T^{2} - 6018 T^{3} + 86511 T^{4} + 1291019 T^{5} - 17948776 T^{6} - 142704563 T^{7} + 3200950152 T^{8} + 10066955091 T^{9} - 386174306271 T^{10} + 417046078782 T^{11} + 36565604004990 T^{12} - 165085514058430 T^{13} - 2122806065999075 T^{14} + 24618965320477429 T^{15} + 41680806306579420 T^{16} - 2234638194195369975 T^{17} + 10624345650292176229 T^{18} +$$$$16\!\cdots\!70$$$$T^{19} -$$$$16\!\cdots\!62$$$$T^{20} -$$$$81\!\cdots\!36$$$$T^{21} +$$$$16\!\cdots\!66$$$$T^{22} +$$$$21\!\cdots\!82$$$$T^{23} -$$$$12\!\cdots\!57$$$$T^{24} +$$$$13\!\cdots\!39$$$$T^{25} +$$$$68\!\cdots\!32$$$$T^{26} -$$$$28\!\cdots\!99$$$$T^{27} -$$$$20\!\cdots\!59$$$$T^{28} +$$$$22\!\cdots\!00$$$$T^{29} -$$$$14\!\cdots\!15$$$$T^{30} -$$$$15\!\cdots\!23$$$$T^{31} +$$$$25\!\cdots\!32$$$$T^{32} +$$$$27\!\cdots\!49$$$$T^{33} -$$$$26\!\cdots\!90$$$$T^{34} +$$$$27\!\cdots\!49$$$$p T^{35} +$$$$25\!\cdots\!32$$$$p^{2} T^{36} -$$$$15\!\cdots\!23$$$$p^{3} T^{37} -$$$$14\!\cdots\!15$$$$p^{4} T^{38} +$$$$22\!\cdots\!00$$$$p^{5} T^{39} -$$$$20\!\cdots\!59$$$$p^{6} T^{40} -$$$$28\!\cdots\!99$$$$p^{7} T^{41} +$$$$68\!\cdots\!32$$$$p^{8} T^{42} +$$$$13\!\cdots\!39$$$$p^{9} T^{43} -$$$$12\!\cdots\!57$$$$p^{10} T^{44} +$$$$21\!\cdots\!82$$$$p^{11} T^{45} +$$$$16\!\cdots\!66$$$$p^{12} T^{46} -$$$$81\!\cdots\!36$$$$p^{13} T^{47} -$$$$16\!\cdots\!62$$$$p^{14} T^{48} +$$$$16\!\cdots\!70$$$$p^{15} T^{49} + 10624345650292176229 p^{16} T^{50} - 2234638194195369975 p^{17} T^{51} + 41680806306579420 p^{18} T^{52} + 24618965320477429 p^{19} T^{53} - 2122806065999075 p^{20} T^{54} - 165085514058430 p^{21} T^{55} + 36565604004990 p^{22} T^{56} + 417046078782 p^{23} T^{57} - 386174306271 p^{24} T^{58} + 10066955091 p^{25} T^{59} + 3200950152 p^{26} T^{60} - 142704563 p^{27} T^{61} - 17948776 p^{28} T^{62} + 1291019 p^{29} T^{63} + 86511 p^{30} T^{64} - 6018 p^{31} T^{65} - 235 p^{32} T^{66} + 23 p^{33} T^{67} + p^{34} T^{68}$$
83 $$1 - 3 T - 765 T^{2} + 2052 T^{3} + 294452 T^{4} - 738714 T^{5} - 75448551 T^{6} + 190659436 T^{7} + 14413404473 T^{8} - 40354993106 T^{9} - 2190499005924 T^{10} + 7453118072043 T^{11} + 277140680017763 T^{12} - 1220476308597524 T^{13} - 30179892702401116 T^{14} + 176472525740382985 T^{15} + 2878507302543348173 T^{16} - 22449973094774520780 T^{17} -$$$$23\!\cdots\!10$$$$T^{18} +$$$$25\!\cdots\!27$$$$T^{19} +$$$$15\!\cdots\!23$$$$T^{20} -$$$$25\!\cdots\!56$$$$T^{21} -$$$$67\!\cdots\!65$$$$T^{22} +$$$$23\!\cdots\!34$$$$T^{23} -$$$$20\!\cdots\!40$$$$T^{24} -$$$$19\!\cdots\!18$$$$T^{25} +$$$$87\!\cdots\!63$$$$T^{26} +$$$$14\!\cdots\!34$$$$T^{27} -$$$$12\!\cdots\!01$$$$T^{28} -$$$$93\!\cdots\!83$$$$T^{29} +$$$$14\!\cdots\!19$$$$T^{30} +$$$$49\!\cdots\!52$$$$T^{31} -$$$$14\!\cdots\!12$$$$T^{32} -$$$$14\!\cdots\!57$$$$T^{33} +$$$$12\!\cdots\!74$$$$T^{34} -$$$$14\!\cdots\!57$$$$p T^{35} -$$$$14\!\cdots\!12$$$$p^{2} T^{36} +$$$$49\!\cdots\!52$$$$p^{3} T^{37} +$$$$14\!\cdots\!19$$$$p^{4} T^{38} -$$$$93\!\cdots\!83$$$$p^{5} T^{39} -$$$$12\!\cdots\!01$$$$p^{6} T^{40} +$$$$14\!\cdots\!34$$$$p^{7} T^{41} +$$$$87\!\cdots\!63$$$$p^{8} T^{42} -$$$$19\!\cdots\!18$$$$p^{9} T^{43} -$$$$20\!\cdots\!40$$$$p^{10} T^{44} +$$$$23\!\cdots\!34$$$$p^{11} T^{45} -$$$$67\!\cdots\!65$$$$p^{12} T^{46} -$$$$25\!\cdots\!56$$$$p^{13} T^{47} +$$$$15\!\cdots\!23$$$$p^{14} T^{48} +$$$$25\!\cdots\!27$$$$p^{15} T^{49} -$$$$23\!\cdots\!10$$$$p^{16} T^{50} - 22449973094774520780 p^{17} T^{51} + 2878507302543348173 p^{18} T^{52} + 176472525740382985 p^{19} T^{53} - 30179892702401116 p^{20} T^{54} - 1220476308597524 p^{21} T^{55} + 277140680017763 p^{22} T^{56} + 7453118072043 p^{23} T^{57} - 2190499005924 p^{24} T^{58} - 40354993106 p^{25} T^{59} + 14413404473 p^{26} T^{60} + 190659436 p^{27} T^{61} - 75448551 p^{28} T^{62} - 738714 p^{29} T^{63} + 294452 p^{30} T^{64} + 2052 p^{31} T^{65} - 765 p^{32} T^{66} - 3 p^{33} T^{67} + p^{34} T^{68}$$
89 $$( 1 - 13 T + 849 T^{2} - 10280 T^{3} + 357515 T^{4} - 4078800 T^{5} + 99977690 T^{6} - 1082978520 T^{7} + 20938623782 T^{8} - 216149909211 T^{9} + 3504302423091 T^{10} - 34484495125622 T^{11} + 487346098090112 T^{12} - 4559714123695934 T^{13} + 57710359417535945 T^{14} - 510633520047265370 T^{15} + 5904316466479830063 T^{16} - 49003025425284858036 T^{17} + 5904316466479830063 p T^{18} - 510633520047265370 p^{2} T^{19} + 57710359417535945 p^{3} T^{20} - 4559714123695934 p^{4} T^{21} + 487346098090112 p^{5} T^{22} - 34484495125622 p^{6} T^{23} + 3504302423091 p^{7} T^{24} - 216149909211 p^{8} T^{25} + 20938623782 p^{9} T^{26} - 1082978520 p^{10} T^{27} + 99977690 p^{11} T^{28} - 4078800 p^{12} T^{29} + 357515 p^{13} T^{30} - 10280 p^{14} T^{31} + 849 p^{15} T^{32} - 13 p^{16} T^{33} + p^{17} T^{34} )^{2}$$
97 $$( 1 - 16 T + 1236 T^{2} - 19199 T^{3} + 751289 T^{4} - 11093335 T^{5} + 297478665 T^{6} - 4113838197 T^{7} + 85668370604 T^{8} - 1099093575250 T^{9} + 18989817868485 T^{10} - 224734587772025 T^{11} + 3348536287754044 T^{12} - 36414703608697428 T^{13} + 479185798567718908 T^{14} - 4771644382489195870 T^{15} + 56303587035684475204 T^{16} -$$$$51\!\cdots\!16$$$$T^{17} + 56303587035684475204 p T^{18} - 4771644382489195870 p^{2} T^{19} + 479185798567718908 p^{3} T^{20} - 36414703608697428 p^{4} T^{21} + 3348536287754044 p^{5} T^{22} - 224734587772025 p^{6} T^{23} + 18989817868485 p^{7} T^{24} - 1099093575250 p^{8} T^{25} + 85668370604 p^{9} T^{26} - 4113838197 p^{10} T^{27} + 297478665 p^{11} T^{28} - 11093335 p^{12} T^{29} + 751289 p^{13} T^{30} - 19199 p^{14} T^{31} + 1236 p^{15} T^{32} - 16 p^{16} T^{33} + p^{17} T^{34} )^{2}$$
\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{68} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}