Dirichlet series
| L(s) = 1 | + 17·2-s + 74·3-s − 1.43e3·4-s + 454·5-s + 1.25e3·6-s − 5.52e3·7-s − 3.29e4·8-s − 7.94e4·9-s + 7.71e3·10-s + 1.52e5·11-s − 1.06e5·12-s + 2.34e4·13-s − 9.39e4·14-s + 3.35e4·15-s + 5.89e5·16-s − 1.34e6·18-s + 1.05e6·19-s − 6.51e5·20-s − 4.08e5·21-s + 2.59e6·22-s + 2.01e6·23-s − 2.43e6·24-s − 8.55e6·25-s + 3.99e5·26-s − 5.41e6·27-s + 7.92e6·28-s + 1.27e7·29-s + ⋯ |
| L(s) = 1 | + 0.751·2-s + 0.527·3-s − 2.80·4-s + 0.324·5-s + 0.396·6-s − 0.869·7-s − 2.84·8-s − 4.03·9-s + 0.244·10-s + 3.14·11-s − 1.47·12-s + 0.227·13-s − 0.653·14-s + 0.171·15-s + 2.24·16-s − 3.03·18-s + 1.85·19-s − 0.909·20-s − 0.458·21-s + 2.36·22-s + 1.49·23-s − 1.49·24-s − 4.37·25-s + 0.171·26-s − 1.96·27-s + 2.43·28-s + 3.35·29-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(17^{24}\right)^{s/2} \, \Gamma_{\C}(s)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(10-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(17^{24}\right)^{s/2} \, \Gamma_{\C}(s+9/2)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(24\) |
| Conductor: | \(17^{24}\) |
| Sign: | $1$ |
| Analytic conductor: | \(1.18256\times 10^{26}\) |
| Root analytic conductor: | \(12.2002\) |
| Motivic weight: | \(9\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((24,\ 17^{24} ,\ ( \ : [9/2]^{12} ),\ 1 )\) |
Particular Values
| \(L(5)\) | \(\approx\) | \(12.13435857\) |
| \(L(\frac12)\) | \(\approx\) | \(12.13435857\) |
| \(L(\frac{11}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 17 | \( 1 \) |
| good | 2 | \( 1 - 17 T + 1723 T^{2} - 5185 p^{2} T^{3} + 418507 p^{2} T^{4} - 1005585 p^{4} T^{5} + 71377811 p^{4} T^{6} - 18003515 p^{9} T^{7} + 2326234983 p^{8} T^{8} - 985731397 p^{12} T^{9} + 64288739221 p^{12} T^{10} - 5886770895 p^{18} T^{11} + 1802241349689 p^{16} T^{12} - 5886770895 p^{27} T^{13} + 64288739221 p^{30} T^{14} - 985731397 p^{39} T^{15} + 2326234983 p^{44} T^{16} - 18003515 p^{54} T^{17} + 71377811 p^{58} T^{18} - 1005585 p^{67} T^{19} + 418507 p^{74} T^{20} - 5185 p^{83} T^{21} + 1723 p^{90} T^{22} - 17 p^{99} T^{23} + p^{108} T^{24} \) |
| 3 | \( 1 - 74 T + 84887 T^{2} - 6738262 T^{3} + 455025904 p^{2} T^{4} - 36052773016 p^{2} T^{5} + 618372215098 p^{5} T^{6} - 49778964393008 p^{5} T^{7} + 6142079587027285 p^{6} T^{8} - 495872075950185656 p^{6} T^{9} + 16945873146376295546 p^{8} T^{10} - \)\(13\!\cdots\!48\)\( p^{8} T^{11} + \)\(13\!\cdots\!39\)\( p^{11} T^{12} - \)\(13\!\cdots\!48\)\( p^{17} T^{13} + 16945873146376295546 p^{26} T^{14} - 495872075950185656 p^{33} T^{15} + 6142079587027285 p^{42} T^{16} - 49778964393008 p^{50} T^{17} + 618372215098 p^{59} T^{18} - 36052773016 p^{65} T^{19} + 455025904 p^{74} T^{20} - 6738262 p^{81} T^{21} + 84887 p^{90} T^{22} - 74 p^{99} T^{23} + p^{108} T^{24} \) | |
| 5 | \( 1 - 454 T + 350409 p^{2} T^{2} - 6160259308 T^{3} + 44389619049286 T^{4} - 30186124625201618 T^{5} + \)\(15\!\cdots\!18\)\( T^{6} - \)\(18\!\cdots\!38\)\( p T^{7} + \)\(35\!\cdots\!87\)\( p^{3} T^{8} - \)\(17\!\cdots\!02\)\( p^{3} T^{9} + \)\(16\!\cdots\!62\)\( p^{4} T^{10} - \)\(11\!\cdots\!82\)\( p^{8} T^{11} + \)\(13\!\cdots\!71\)\( p^{6} T^{12} - \)\(11\!\cdots\!82\)\( p^{17} T^{13} + \)\(16\!\cdots\!62\)\( p^{22} T^{14} - \)\(17\!\cdots\!02\)\( p^{30} T^{15} + \)\(35\!\cdots\!87\)\( p^{39} T^{16} - \)\(18\!\cdots\!38\)\( p^{46} T^{17} + \)\(15\!\cdots\!18\)\( p^{54} T^{18} - 30186124625201618 p^{63} T^{19} + 44389619049286 p^{72} T^{20} - 6160259308 p^{81} T^{21} + 350409 p^{92} T^{22} - 454 p^{99} T^{23} + p^{108} T^{24} \) | |
| 7 | \( 1 + 5524 T + 165134905 T^{2} + 882767242850 T^{3} + 13909888437038584 T^{4} + 11142674204726191306 p T^{5} + \)\(17\!\cdots\!78\)\( p^{2} T^{6} + \)\(15\!\cdots\!22\)\( p^{3} T^{7} + \)\(19\!\cdots\!25\)\( p^{4} T^{8} + \)\(16\!\cdots\!66\)\( p^{5} T^{9} + \)\(26\!\cdots\!74\)\( p^{7} T^{10} + \)\(22\!\cdots\!38\)\( p^{8} T^{11} + \)\(16\!\cdots\!09\)\( p^{8} T^{12} + \)\(22\!\cdots\!38\)\( p^{17} T^{13} + \)\(26\!\cdots\!74\)\( p^{25} T^{14} + \)\(16\!\cdots\!66\)\( p^{32} T^{15} + \)\(19\!\cdots\!25\)\( p^{40} T^{16} + \)\(15\!\cdots\!22\)\( p^{48} T^{17} + \)\(17\!\cdots\!78\)\( p^{56} T^{18} + 11142674204726191306 p^{64} T^{19} + 13909888437038584 p^{72} T^{20} + 882767242850 p^{81} T^{21} + 165134905 p^{90} T^{22} + 5524 p^{99} T^{23} + p^{108} T^{24} \) | |
| 11 | \( 1 - 152886 T + 23785441164 T^{2} - 2293884428585424 T^{3} + \)\(21\!\cdots\!91\)\( T^{4} - \)\(16\!\cdots\!40\)\( T^{5} + \)\(11\!\cdots\!52\)\( T^{6} - \)\(71\!\cdots\!94\)\( T^{7} + \)\(43\!\cdots\!90\)\( T^{8} - \)\(21\!\cdots\!58\)\( p T^{9} + \)\(12\!\cdots\!80\)\( T^{10} - \)\(61\!\cdots\!88\)\( T^{11} + \)\(31\!\cdots\!39\)\( T^{12} - \)\(61\!\cdots\!88\)\( p^{9} T^{13} + \)\(12\!\cdots\!80\)\( p^{18} T^{14} - \)\(21\!\cdots\!58\)\( p^{28} T^{15} + \)\(43\!\cdots\!90\)\( p^{36} T^{16} - \)\(71\!\cdots\!94\)\( p^{45} T^{17} + \)\(11\!\cdots\!52\)\( p^{54} T^{18} - \)\(16\!\cdots\!40\)\( p^{63} T^{19} + \)\(21\!\cdots\!91\)\( p^{72} T^{20} - 2293884428585424 p^{81} T^{21} + 23785441164 p^{90} T^{22} - 152886 p^{99} T^{23} + p^{108} T^{24} \) | |
| 13 | \( 1 - 1806 p T + 61881417953 T^{2} - 2837972571329980 T^{3} + \)\(19\!\cdots\!98\)\( T^{4} - \)\(12\!\cdots\!90\)\( T^{5} + \)\(32\!\cdots\!90\)\( p T^{6} - \)\(31\!\cdots\!34\)\( T^{7} + \)\(71\!\cdots\!51\)\( T^{8} - \)\(55\!\cdots\!10\)\( T^{9} + \)\(10\!\cdots\!66\)\( T^{10} - \)\(74\!\cdots\!06\)\( T^{11} + \)\(90\!\cdots\!35\)\( p T^{12} - \)\(74\!\cdots\!06\)\( p^{9} T^{13} + \)\(10\!\cdots\!66\)\( p^{18} T^{14} - \)\(55\!\cdots\!10\)\( p^{27} T^{15} + \)\(71\!\cdots\!51\)\( p^{36} T^{16} - \)\(31\!\cdots\!34\)\( p^{45} T^{17} + \)\(32\!\cdots\!90\)\( p^{55} T^{18} - \)\(12\!\cdots\!90\)\( p^{63} T^{19} + \)\(19\!\cdots\!98\)\( p^{72} T^{20} - 2837972571329980 p^{81} T^{21} + 61881417953 p^{90} T^{22} - 1806 p^{100} T^{23} + p^{108} T^{24} \) | |
| 19 | \( 1 - 1053982 T + 2607497196559 T^{2} - 2267614599389735442 T^{3} + \)\(31\!\cdots\!40\)\( T^{4} - \)\(23\!\cdots\!96\)\( T^{5} + \)\(24\!\cdots\!62\)\( T^{6} - \)\(15\!\cdots\!96\)\( T^{7} + \)\(13\!\cdots\!09\)\( T^{8} - \)\(77\!\cdots\!68\)\( T^{9} + \)\(59\!\cdots\!82\)\( T^{10} - \)\(30\!\cdots\!24\)\( T^{11} + \)\(20\!\cdots\!77\)\( T^{12} - \)\(30\!\cdots\!24\)\( p^{9} T^{13} + \)\(59\!\cdots\!82\)\( p^{18} T^{14} - \)\(77\!\cdots\!68\)\( p^{27} T^{15} + \)\(13\!\cdots\!09\)\( p^{36} T^{16} - \)\(15\!\cdots\!96\)\( p^{45} T^{17} + \)\(24\!\cdots\!62\)\( p^{54} T^{18} - \)\(23\!\cdots\!96\)\( p^{63} T^{19} + \)\(31\!\cdots\!40\)\( p^{72} T^{20} - 2267614599389735442 p^{81} T^{21} + 2607497196559 p^{90} T^{22} - 1053982 p^{99} T^{23} + p^{108} T^{24} \) | |
| 23 | \( 1 - 2012428 T + 12428795484981 T^{2} - 18669977654521570618 T^{3} + \)\(73\!\cdots\!84\)\( T^{4} - \)\(88\!\cdots\!98\)\( T^{5} + \)\(12\!\cdots\!46\)\( p T^{6} - \)\(27\!\cdots\!22\)\( T^{7} + \)\(78\!\cdots\!73\)\( T^{8} - \)\(65\!\cdots\!14\)\( T^{9} + \)\(17\!\cdots\!98\)\( T^{10} - \)\(13\!\cdots\!74\)\( T^{11} + \)\(34\!\cdots\!93\)\( T^{12} - \)\(13\!\cdots\!74\)\( p^{9} T^{13} + \)\(17\!\cdots\!98\)\( p^{18} T^{14} - \)\(65\!\cdots\!14\)\( p^{27} T^{15} + \)\(78\!\cdots\!73\)\( p^{36} T^{16} - \)\(27\!\cdots\!22\)\( p^{45} T^{17} + \)\(12\!\cdots\!46\)\( p^{55} T^{18} - \)\(88\!\cdots\!98\)\( p^{63} T^{19} + \)\(73\!\cdots\!84\)\( p^{72} T^{20} - 18669977654521570618 p^{81} T^{21} + 12428795484981 p^{90} T^{22} - 2012428 p^{99} T^{23} + p^{108} T^{24} \) | |
| 29 | \( 1 - 12772842 T + 154722938298917 T^{2} - \)\(13\!\cdots\!04\)\( T^{3} + \)\(11\!\cdots\!38\)\( T^{4} - \)\(77\!\cdots\!38\)\( T^{5} + \)\(48\!\cdots\!02\)\( T^{6} - \)\(27\!\cdots\!94\)\( T^{7} + \)\(14\!\cdots\!35\)\( T^{8} - \)\(71\!\cdots\!06\)\( T^{9} + \)\(32\!\cdots\!02\)\( T^{10} - \)\(13\!\cdots\!10\)\( T^{11} + \)\(18\!\cdots\!03\)\( p T^{12} - \)\(13\!\cdots\!10\)\( p^{9} T^{13} + \)\(32\!\cdots\!02\)\( p^{18} T^{14} - \)\(71\!\cdots\!06\)\( p^{27} T^{15} + \)\(14\!\cdots\!35\)\( p^{36} T^{16} - \)\(27\!\cdots\!94\)\( p^{45} T^{17} + \)\(48\!\cdots\!02\)\( p^{54} T^{18} - \)\(77\!\cdots\!38\)\( p^{63} T^{19} + \)\(11\!\cdots\!38\)\( p^{72} T^{20} - \)\(13\!\cdots\!04\)\( p^{81} T^{21} + 154722938298917 p^{90} T^{22} - 12772842 p^{99} T^{23} + p^{108} T^{24} \) | |
| 31 | \( 1 - 5535814 T + 193517307747004 T^{2} - \)\(92\!\cdots\!56\)\( T^{3} + \)\(18\!\cdots\!75\)\( T^{4} - \)\(73\!\cdots\!16\)\( T^{5} + \)\(10\!\cdots\!20\)\( T^{6} - \)\(36\!\cdots\!94\)\( T^{7} + \)\(45\!\cdots\!18\)\( T^{8} - \)\(13\!\cdots\!38\)\( T^{9} + \)\(15\!\cdots\!04\)\( T^{10} - \)\(13\!\cdots\!36\)\( p T^{11} + \)\(44\!\cdots\!23\)\( T^{12} - \)\(13\!\cdots\!36\)\( p^{10} T^{13} + \)\(15\!\cdots\!04\)\( p^{18} T^{14} - \)\(13\!\cdots\!38\)\( p^{27} T^{15} + \)\(45\!\cdots\!18\)\( p^{36} T^{16} - \)\(36\!\cdots\!94\)\( p^{45} T^{17} + \)\(10\!\cdots\!20\)\( p^{54} T^{18} - \)\(73\!\cdots\!16\)\( p^{63} T^{19} + \)\(18\!\cdots\!75\)\( p^{72} T^{20} - \)\(92\!\cdots\!56\)\( p^{81} T^{21} + 193517307747004 p^{90} T^{22} - 5535814 p^{99} T^{23} + p^{108} T^{24} \) | |
| 37 | \( 1 - 1352872 T + 788642816062120 T^{2} - \)\(20\!\cdots\!16\)\( T^{3} + \)\(32\!\cdots\!78\)\( T^{4} - \)\(11\!\cdots\!52\)\( T^{5} + \)\(90\!\cdots\!56\)\( T^{6} - \)\(38\!\cdots\!68\)\( T^{7} + \)\(19\!\cdots\!87\)\( T^{8} - \)\(86\!\cdots\!44\)\( T^{9} + \)\(33\!\cdots\!20\)\( T^{10} - \)\(14\!\cdots\!16\)\( T^{11} + \)\(48\!\cdots\!36\)\( T^{12} - \)\(14\!\cdots\!16\)\( p^{9} T^{13} + \)\(33\!\cdots\!20\)\( p^{18} T^{14} - \)\(86\!\cdots\!44\)\( p^{27} T^{15} + \)\(19\!\cdots\!87\)\( p^{36} T^{16} - \)\(38\!\cdots\!68\)\( p^{45} T^{17} + \)\(90\!\cdots\!56\)\( p^{54} T^{18} - \)\(11\!\cdots\!52\)\( p^{63} T^{19} + \)\(32\!\cdots\!78\)\( p^{72} T^{20} - \)\(20\!\cdots\!16\)\( p^{81} T^{21} + 788642816062120 p^{90} T^{22} - 1352872 p^{99} T^{23} + p^{108} T^{24} \) | |
| 41 | \( 1 - 40941240 T + 2260742772214532 T^{2} - \)\(63\!\cdots\!88\)\( T^{3} + \)\(20\!\cdots\!78\)\( T^{4} - \)\(48\!\cdots\!76\)\( T^{5} + \)\(12\!\cdots\!76\)\( T^{6} - \)\(27\!\cdots\!72\)\( T^{7} + \)\(63\!\cdots\!39\)\( T^{8} - \)\(12\!\cdots\!48\)\( T^{9} + \)\(25\!\cdots\!24\)\( T^{10} - \)\(46\!\cdots\!44\)\( T^{11} + \)\(90\!\cdots\!84\)\( T^{12} - \)\(46\!\cdots\!44\)\( p^{9} T^{13} + \)\(25\!\cdots\!24\)\( p^{18} T^{14} - \)\(12\!\cdots\!48\)\( p^{27} T^{15} + \)\(63\!\cdots\!39\)\( p^{36} T^{16} - \)\(27\!\cdots\!72\)\( p^{45} T^{17} + \)\(12\!\cdots\!76\)\( p^{54} T^{18} - \)\(48\!\cdots\!76\)\( p^{63} T^{19} + \)\(20\!\cdots\!78\)\( p^{72} T^{20} - \)\(63\!\cdots\!88\)\( p^{81} T^{21} + 2260742772214532 p^{90} T^{22} - 40941240 p^{99} T^{23} + p^{108} T^{24} \) | |
| 43 | \( 1 + 14142490 T + 3341279528566672 T^{2} + \)\(53\!\cdots\!48\)\( T^{3} + \)\(56\!\cdots\!95\)\( T^{4} + \)\(99\!\cdots\!68\)\( T^{5} + \)\(63\!\cdots\!96\)\( T^{6} + \)\(12\!\cdots\!38\)\( T^{7} + \)\(53\!\cdots\!14\)\( T^{8} + \)\(10\!\cdots\!34\)\( T^{9} + \)\(35\!\cdots\!92\)\( T^{10} + \)\(68\!\cdots\!12\)\( T^{11} + \)\(19\!\cdots\!47\)\( T^{12} + \)\(68\!\cdots\!12\)\( p^{9} T^{13} + \)\(35\!\cdots\!92\)\( p^{18} T^{14} + \)\(10\!\cdots\!34\)\( p^{27} T^{15} + \)\(53\!\cdots\!14\)\( p^{36} T^{16} + \)\(12\!\cdots\!38\)\( p^{45} T^{17} + \)\(63\!\cdots\!96\)\( p^{54} T^{18} + \)\(99\!\cdots\!68\)\( p^{63} T^{19} + \)\(56\!\cdots\!95\)\( p^{72} T^{20} + \)\(53\!\cdots\!48\)\( p^{81} T^{21} + 3341279528566672 p^{90} T^{22} + 14142490 p^{99} T^{23} + p^{108} T^{24} \) | |
| 47 | \( 1 - 133558002 T + 15844054411405944 T^{2} - \)\(12\!\cdots\!32\)\( T^{3} + \)\(89\!\cdots\!51\)\( T^{4} - \)\(52\!\cdots\!88\)\( T^{5} + \)\(29\!\cdots\!32\)\( T^{6} - \)\(14\!\cdots\!18\)\( T^{7} + \)\(65\!\cdots\!98\)\( T^{8} - \)\(27\!\cdots\!98\)\( T^{9} + \)\(10\!\cdots\!80\)\( T^{10} - \)\(39\!\cdots\!04\)\( T^{11} + \)\(13\!\cdots\!79\)\( T^{12} - \)\(39\!\cdots\!04\)\( p^{9} T^{13} + \)\(10\!\cdots\!80\)\( p^{18} T^{14} - \)\(27\!\cdots\!98\)\( p^{27} T^{15} + \)\(65\!\cdots\!98\)\( p^{36} T^{16} - \)\(14\!\cdots\!18\)\( p^{45} T^{17} + \)\(29\!\cdots\!32\)\( p^{54} T^{18} - \)\(52\!\cdots\!88\)\( p^{63} T^{19} + \)\(89\!\cdots\!51\)\( p^{72} T^{20} - \)\(12\!\cdots\!32\)\( p^{81} T^{21} + 15844054411405944 p^{90} T^{22} - 133558002 p^{99} T^{23} + p^{108} T^{24} \) | |
| 53 | \( 1 + 299319258 T + 56085124359636549 T^{2} + \)\(75\!\cdots\!00\)\( T^{3} + \)\(82\!\cdots\!98\)\( T^{4} + \)\(77\!\cdots\!18\)\( T^{5} + \)\(65\!\cdots\!18\)\( T^{6} + \)\(51\!\cdots\!54\)\( T^{7} + \)\(37\!\cdots\!75\)\( T^{8} + \)\(48\!\cdots\!66\)\( p T^{9} + \)\(16\!\cdots\!06\)\( T^{10} + \)\(10\!\cdots\!02\)\( T^{11} + \)\(59\!\cdots\!43\)\( T^{12} + \)\(10\!\cdots\!02\)\( p^{9} T^{13} + \)\(16\!\cdots\!06\)\( p^{18} T^{14} + \)\(48\!\cdots\!66\)\( p^{28} T^{15} + \)\(37\!\cdots\!75\)\( p^{36} T^{16} + \)\(51\!\cdots\!54\)\( p^{45} T^{17} + \)\(65\!\cdots\!18\)\( p^{54} T^{18} + \)\(77\!\cdots\!18\)\( p^{63} T^{19} + \)\(82\!\cdots\!98\)\( p^{72} T^{20} + \)\(75\!\cdots\!00\)\( p^{81} T^{21} + 56085124359636549 p^{90} T^{22} + 299319258 p^{99} T^{23} + p^{108} T^{24} \) | |
| 59 | \( 1 + 106431852 T + 71971344576985844 T^{2} + \)\(77\!\cdots\!72\)\( T^{3} + \)\(25\!\cdots\!54\)\( T^{4} + \)\(26\!\cdots\!48\)\( T^{5} + \)\(59\!\cdots\!80\)\( T^{6} + \)\(58\!\cdots\!72\)\( T^{7} + \)\(99\!\cdots\!11\)\( T^{8} + \)\(90\!\cdots\!60\)\( T^{9} + \)\(12\!\cdots\!56\)\( T^{10} + \)\(10\!\cdots\!28\)\( T^{11} + \)\(12\!\cdots\!96\)\( T^{12} + \)\(10\!\cdots\!28\)\( p^{9} T^{13} + \)\(12\!\cdots\!56\)\( p^{18} T^{14} + \)\(90\!\cdots\!60\)\( p^{27} T^{15} + \)\(99\!\cdots\!11\)\( p^{36} T^{16} + \)\(58\!\cdots\!72\)\( p^{45} T^{17} + \)\(59\!\cdots\!80\)\( p^{54} T^{18} + \)\(26\!\cdots\!48\)\( p^{63} T^{19} + \)\(25\!\cdots\!54\)\( p^{72} T^{20} + \)\(77\!\cdots\!72\)\( p^{81} T^{21} + 71971344576985844 p^{90} T^{22} + 106431852 p^{99} T^{23} + p^{108} T^{24} \) | |
| 61 | \( 1 - 262041240 T + 87585937447272614 T^{2} - \)\(16\!\cdots\!44\)\( T^{3} + \)\(31\!\cdots\!81\)\( T^{4} - \)\(44\!\cdots\!84\)\( T^{5} + \)\(60\!\cdots\!30\)\( T^{6} - \)\(64\!\cdots\!80\)\( T^{7} + \)\(10\!\cdots\!18\)\( p T^{8} - \)\(46\!\cdots\!08\)\( T^{9} + \)\(33\!\cdots\!90\)\( T^{10} - \)\(86\!\cdots\!32\)\( T^{11} + \)\(89\!\cdots\!41\)\( T^{12} - \)\(86\!\cdots\!32\)\( p^{9} T^{13} + \)\(33\!\cdots\!90\)\( p^{18} T^{14} - \)\(46\!\cdots\!08\)\( p^{27} T^{15} + \)\(10\!\cdots\!18\)\( p^{37} T^{16} - \)\(64\!\cdots\!80\)\( p^{45} T^{17} + \)\(60\!\cdots\!30\)\( p^{54} T^{18} - \)\(44\!\cdots\!84\)\( p^{63} T^{19} + \)\(31\!\cdots\!81\)\( p^{72} T^{20} - \)\(16\!\cdots\!44\)\( p^{81} T^{21} + 87585937447272614 p^{90} T^{22} - 262041240 p^{99} T^{23} + p^{108} T^{24} \) | |
| 67 | \( 1 + 489635100 T + 297223536400185072 T^{2} + \)\(10\!\cdots\!76\)\( T^{3} + \)\(39\!\cdots\!78\)\( T^{4} + \)\(11\!\cdots\!48\)\( T^{5} + \)\(31\!\cdots\!96\)\( T^{6} + \)\(74\!\cdots\!40\)\( T^{7} + \)\(17\!\cdots\!87\)\( T^{8} + \)\(35\!\cdots\!72\)\( T^{9} + \)\(71\!\cdots\!84\)\( T^{10} + \)\(12\!\cdots\!68\)\( T^{11} + \)\(22\!\cdots\!76\)\( T^{12} + \)\(12\!\cdots\!68\)\( p^{9} T^{13} + \)\(71\!\cdots\!84\)\( p^{18} T^{14} + \)\(35\!\cdots\!72\)\( p^{27} T^{15} + \)\(17\!\cdots\!87\)\( p^{36} T^{16} + \)\(74\!\cdots\!40\)\( p^{45} T^{17} + \)\(31\!\cdots\!96\)\( p^{54} T^{18} + \)\(11\!\cdots\!48\)\( p^{63} T^{19} + \)\(39\!\cdots\!78\)\( p^{72} T^{20} + \)\(10\!\cdots\!76\)\( p^{81} T^{21} + 297223536400185072 p^{90} T^{22} + 489635100 p^{99} T^{23} + p^{108} T^{24} \) | |
| 71 | \( 1 - 204290852 T + 330637037076443956 T^{2} - \)\(30\!\cdots\!20\)\( T^{3} + \)\(43\!\cdots\!98\)\( T^{4} + \)\(10\!\cdots\!20\)\( T^{5} + \)\(32\!\cdots\!36\)\( T^{6} + \)\(46\!\cdots\!76\)\( T^{7} + \)\(21\!\cdots\!47\)\( T^{8} + \)\(38\!\cdots\!80\)\( T^{9} + \)\(14\!\cdots\!04\)\( T^{10} + \)\(18\!\cdots\!64\)\( T^{11} + \)\(80\!\cdots\!80\)\( T^{12} + \)\(18\!\cdots\!64\)\( p^{9} T^{13} + \)\(14\!\cdots\!04\)\( p^{18} T^{14} + \)\(38\!\cdots\!80\)\( p^{27} T^{15} + \)\(21\!\cdots\!47\)\( p^{36} T^{16} + \)\(46\!\cdots\!76\)\( p^{45} T^{17} + \)\(32\!\cdots\!36\)\( p^{54} T^{18} + \)\(10\!\cdots\!20\)\( p^{63} T^{19} + \)\(43\!\cdots\!98\)\( p^{72} T^{20} - \)\(30\!\cdots\!20\)\( p^{81} T^{21} + 330637037076443956 p^{90} T^{22} - 204290852 p^{99} T^{23} + p^{108} T^{24} \) | |
| 73 | \( 1 - 673538852 T + 614230692079380650 T^{2} - \)\(30\!\cdots\!16\)\( T^{3} + \)\(17\!\cdots\!97\)\( T^{4} - \)\(69\!\cdots\!32\)\( T^{5} + \)\(40\!\cdots\!94\)\( p T^{6} - \)\(10\!\cdots\!28\)\( T^{7} + \)\(35\!\cdots\!78\)\( T^{8} - \)\(10\!\cdots\!76\)\( T^{9} + \)\(31\!\cdots\!82\)\( T^{10} - \)\(81\!\cdots\!88\)\( T^{11} + \)\(21\!\cdots\!61\)\( T^{12} - \)\(81\!\cdots\!88\)\( p^{9} T^{13} + \)\(31\!\cdots\!82\)\( p^{18} T^{14} - \)\(10\!\cdots\!76\)\( p^{27} T^{15} + \)\(35\!\cdots\!78\)\( p^{36} T^{16} - \)\(10\!\cdots\!28\)\( p^{45} T^{17} + \)\(40\!\cdots\!94\)\( p^{55} T^{18} - \)\(69\!\cdots\!32\)\( p^{63} T^{19} + \)\(17\!\cdots\!97\)\( p^{72} T^{20} - \)\(30\!\cdots\!16\)\( p^{81} T^{21} + 614230692079380650 p^{90} T^{22} - 673538852 p^{99} T^{23} + p^{108} T^{24} \) | |
| 79 | \( 1 - 434002980 T + 1054548728246817636 T^{2} - \)\(45\!\cdots\!04\)\( T^{3} + \)\(54\!\cdots\!50\)\( T^{4} - \)\(22\!\cdots\!96\)\( T^{5} + \)\(18\!\cdots\!64\)\( T^{6} - \)\(69\!\cdots\!52\)\( T^{7} + \)\(44\!\cdots\!51\)\( T^{8} - \)\(15\!\cdots\!44\)\( T^{9} + \)\(79\!\cdots\!24\)\( T^{10} - \)\(24\!\cdots\!68\)\( T^{11} + \)\(10\!\cdots\!00\)\( T^{12} - \)\(24\!\cdots\!68\)\( p^{9} T^{13} + \)\(79\!\cdots\!24\)\( p^{18} T^{14} - \)\(15\!\cdots\!44\)\( p^{27} T^{15} + \)\(44\!\cdots\!51\)\( p^{36} T^{16} - \)\(69\!\cdots\!52\)\( p^{45} T^{17} + \)\(18\!\cdots\!64\)\( p^{54} T^{18} - \)\(22\!\cdots\!96\)\( p^{63} T^{19} + \)\(54\!\cdots\!50\)\( p^{72} T^{20} - \)\(45\!\cdots\!04\)\( p^{81} T^{21} + 1054548728246817636 p^{90} T^{22} - 434002980 p^{99} T^{23} + p^{108} T^{24} \) | |
| 83 | \( 1 - 411781442 T + 1271239689254403715 T^{2} - \)\(37\!\cdots\!18\)\( T^{3} + \)\(77\!\cdots\!72\)\( T^{4} - \)\(16\!\cdots\!24\)\( T^{5} + \)\(30\!\cdots\!78\)\( T^{6} - \)\(43\!\cdots\!20\)\( T^{7} + \)\(91\!\cdots\!85\)\( T^{8} - \)\(84\!\cdots\!20\)\( T^{9} + \)\(22\!\cdots\!42\)\( T^{10} - \)\(14\!\cdots\!20\)\( T^{11} + \)\(44\!\cdots\!29\)\( T^{12} - \)\(14\!\cdots\!20\)\( p^{9} T^{13} + \)\(22\!\cdots\!42\)\( p^{18} T^{14} - \)\(84\!\cdots\!20\)\( p^{27} T^{15} + \)\(91\!\cdots\!85\)\( p^{36} T^{16} - \)\(43\!\cdots\!20\)\( p^{45} T^{17} + \)\(30\!\cdots\!78\)\( p^{54} T^{18} - \)\(16\!\cdots\!24\)\( p^{63} T^{19} + \)\(77\!\cdots\!72\)\( p^{72} T^{20} - \)\(37\!\cdots\!18\)\( p^{81} T^{21} + 1271239689254403715 p^{90} T^{22} - 411781442 p^{99} T^{23} + p^{108} T^{24} \) | |
| 89 | \( 1 - 911678128 T + 1725686634015411060 T^{2} - \)\(12\!\cdots\!56\)\( T^{3} + \)\(12\!\cdots\!06\)\( T^{4} - \)\(74\!\cdots\!64\)\( T^{5} + \)\(57\!\cdots\!80\)\( T^{6} - \)\(29\!\cdots\!12\)\( T^{7} + \)\(21\!\cdots\!71\)\( T^{8} - \)\(11\!\cdots\!52\)\( T^{9} + \)\(87\!\cdots\!48\)\( T^{10} - \)\(44\!\cdots\!60\)\( T^{11} + \)\(33\!\cdots\!76\)\( T^{12} - \)\(44\!\cdots\!60\)\( p^{9} T^{13} + \)\(87\!\cdots\!48\)\( p^{18} T^{14} - \)\(11\!\cdots\!52\)\( p^{27} T^{15} + \)\(21\!\cdots\!71\)\( p^{36} T^{16} - \)\(29\!\cdots\!12\)\( p^{45} T^{17} + \)\(57\!\cdots\!80\)\( p^{54} T^{18} - \)\(74\!\cdots\!64\)\( p^{63} T^{19} + \)\(12\!\cdots\!06\)\( p^{72} T^{20} - \)\(12\!\cdots\!56\)\( p^{81} T^{21} + 1725686634015411060 p^{90} T^{22} - 911678128 p^{99} T^{23} + p^{108} T^{24} \) | |
| 97 | \( 1 + 3589270998 T + 12504456568339410401 T^{2} + \)\(28\!\cdots\!96\)\( T^{3} + \)\(58\!\cdots\!38\)\( T^{4} + \)\(98\!\cdots\!14\)\( T^{5} + \)\(15\!\cdots\!98\)\( T^{6} + \)\(21\!\cdots\!74\)\( T^{7} + \)\(26\!\cdots\!31\)\( T^{8} + \)\(30\!\cdots\!82\)\( T^{9} + \)\(32\!\cdots\!78\)\( T^{10} + \)\(31\!\cdots\!14\)\( T^{11} + \)\(28\!\cdots\!23\)\( T^{12} + \)\(31\!\cdots\!14\)\( p^{9} T^{13} + \)\(32\!\cdots\!78\)\( p^{18} T^{14} + \)\(30\!\cdots\!82\)\( p^{27} T^{15} + \)\(26\!\cdots\!31\)\( p^{36} T^{16} + \)\(21\!\cdots\!74\)\( p^{45} T^{17} + \)\(15\!\cdots\!98\)\( p^{54} T^{18} + \)\(98\!\cdots\!14\)\( p^{63} T^{19} + \)\(58\!\cdots\!38\)\( p^{72} T^{20} + \)\(28\!\cdots\!96\)\( p^{81} T^{21} + 12504456568339410401 p^{90} T^{22} + 3589270998 p^{99} T^{23} + p^{108} T^{24} \) | |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{24} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−2.72085810066388155849385190056, −2.43534120907786815378724907895, −2.31698945909828531913130976166, −2.29824153441885054590022281075, −2.23145957941364445973217002651, −2.01824355766585756048850580826, −1.84768424667367612188586875352, −1.80089348576340131622034255675, −1.78658569781039480771359255043, −1.67779609729096108611017002680, −1.33596135052880507983714748529, −1.29854803378919387516797164951, −1.25083884508154053422111071527, −1.21468480177421572742975582408, −1.19329106913217343149694055413, −0.866510849237946098742316376372, −0.72268804874247738186288569704, −0.71764108539507397137851272352, −0.56729548320981251046318536101, −0.56479054346344194627271203561, −0.50347955364353948814934858275, −0.39899556401412312259355188628, −0.21176156859672893335134519412, −0.19391774298699750913810889467, −0.17711199172164413433163902473, 0.17711199172164413433163902473, 0.19391774298699750913810889467, 0.21176156859672893335134519412, 0.39899556401412312259355188628, 0.50347955364353948814934858275, 0.56479054346344194627271203561, 0.56729548320981251046318536101, 0.71764108539507397137851272352, 0.72268804874247738186288569704, 0.866510849237946098742316376372, 1.19329106913217343149694055413, 1.21468480177421572742975582408, 1.25083884508154053422111071527, 1.29854803378919387516797164951, 1.33596135052880507983714748529, 1.67779609729096108611017002680, 1.78658569781039480771359255043, 1.80089348576340131622034255675, 1.84768424667367612188586875352, 2.01824355766585756048850580826, 2.23145957941364445973217002651, 2.29824153441885054590022281075, 2.31698945909828531913130976166, 2.43534120907786815378724907895, 2.72085810066388155849385190056
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.