Properties

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

Origins

Origins of factors

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Normalization:  

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 \( ( 1 - 16 T + 1236 T^{2} - 19199 T^{3} + 751289 T^{4} - 11093335 T^{5} + 297478665 T^{6} - 4113838197 T^{7} + 85668370604 T^{8} - 1099093575250 T^{9} + 18989817868485 T^{10} - 224734587772025 T^{11} + 3348536287754044 T^{12} - 36414703608697428 T^{13} + 479185798567718908 T^{14} - 4771644382489195870 T^{15} + 56303587035684475204 T^{16} - \)\(51\!\cdots\!16\)\( T^{17} + 56303587035684475204 p T^{18} - 4771644382489195870 p^{2} T^{19} + 479185798567718908 p^{3} T^{20} - 36414703608697428 p^{4} T^{21} + 3348536287754044 p^{5} T^{22} - 224734587772025 p^{6} T^{23} + 18989817868485 p^{7} T^{24} - 1099093575250 p^{8} T^{25} + 85668370604 p^{9} T^{26} - 4113838197 p^{10} T^{27} + 297478665 p^{11} T^{28} - 11093335 p^{12} T^{29} + 751289 p^{13} T^{30} - 19199 p^{14} T^{31} + 1236 p^{15} T^{32} - 16 p^{16} T^{33} + p^{17} T^{34} )^{2} \)
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\[\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{68} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}\]

Imaginary part of the first few zeros on the critical line

−1.94854297802465522180152401582, −1.89419886146785648365660731701, −1.88763374754740957359149658232, −1.79617243844052592378229575332, −1.65580562950103542528505491487, −1.64114733283905714613143450961, −1.60362165562047204490384751677, −1.58885843782493132716816458317, −1.43658693629296870248070333865, −1.37034640457333119587453702330, −1.33954784498907621115743387041, −1.29146094204099107563086214372, −1.07438691195020510448966930793, −0.995734086912398261125855202002, −0.994108209469360806341320947438, −0.870217187233968550440883429473, −0.794300006285590677471971039458, −0.72442242168256544961592726379, −0.67114498117019022897724990599, −0.64270724016019112699910672773, −0.49244628724699744124650724109, −0.47981980884635773509214178351, −0.37558959472461373583541011086, −0.094497343625998001314373657846, −0.081023655658243178625881327101, 0.081023655658243178625881327101, 0.094497343625998001314373657846, 0.37558959472461373583541011086, 0.47981980884635773509214178351, 0.49244628724699744124650724109, 0.64270724016019112699910672773, 0.67114498117019022897724990599, 0.72442242168256544961592726379, 0.794300006285590677471971039458, 0.870217187233968550440883429473, 0.994108209469360806341320947438, 0.995734086912398261125855202002, 1.07438691195020510448966930793, 1.29146094204099107563086214372, 1.33954784498907621115743387041, 1.37034640457333119587453702330, 1.43658693629296870248070333865, 1.58885843782493132716816458317, 1.60362165562047204490384751677, 1.64114733283905714613143450961, 1.65580562950103542528505491487, 1.79617243844052592378229575332, 1.88763374754740957359149658232, 1.89419886146785648365660731701, 1.94854297802465522180152401582

Graph of the $Z$-function along the critical line

Plot not available for L-functions of degree greater than 10.