Dirichlet series
| L(s) = 1 | − 32·2-s − 251·3-s − 1.45e3·4-s − 2.15e3·5-s + 8.03e3·6-s − 1.25e4·7-s + 5.60e4·8-s − 6.18e4·9-s + 6.90e4·10-s − 1.12e5·11-s + 3.64e5·12-s + 7.12e3·13-s + 4.02e5·14-s + 5.41e5·15-s + 6.94e5·16-s − 8.90e5·17-s + 1.98e6·18-s − 1.43e6·19-s + 3.13e6·20-s + 3.15e6·21-s + 3.59e6·22-s − 2.56e6·23-s − 1.40e7·24-s − 1.12e7·25-s − 2.28e5·26-s + 1.90e7·27-s + 1.82e7·28-s + ⋯ |
| L(s) = 1 | − 1.41·2-s − 1.78·3-s − 2.83·4-s − 1.54·5-s + 2.53·6-s − 1.97·7-s + 4.83·8-s − 3.14·9-s + 2.18·10-s − 2.31·11-s + 5.07·12-s + 0.0692·13-s + 2.79·14-s + 2.76·15-s + 2.65·16-s − 2.58·17-s + 4.44·18-s − 2.52·19-s + 4.37·20-s + 3.54·21-s + 3.27·22-s − 1.91·23-s − 8.65·24-s − 5.77·25-s − 0.0979·26-s + 6.90·27-s + 5.61·28-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(37^{13}\right)^{s/2} \, \Gamma_{\C}(s)^{13} \, L(s)\cr=\mathstrut & -\,\Lambda(10-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(37^{13}\right)^{s/2} \, \Gamma_{\C}(s+9/2)^{13} \, L(s)\cr=\mathstrut & -\,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(26\) |
| Conductor: | \(37^{13}\) |
| Sign: | $-1$ |
| Analytic conductor: | \(4.37027\times 10^{16}\) |
| Root analytic conductor: | \(4.36535\) |
| Motivic weight: | \(9\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(13\) |
| Selberg data: | \((26,\ 37^{13} ,\ ( \ : [9/2]^{13} ),\ -1 )\) |
Particular Values
| \(L(5)\) | \(=\) | \(0\) |
| \(L(\frac12)\) | \(=\) | \(0\) |
| \(L(\frac{11}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 37 | \( ( 1 + p^{4} T )^{13} \) |
| good | 2 | \( 1 + p^{5} T + 2475 T^{2} + 34807 p T^{3} + 1665675 p T^{4} + 11044843 p^{3} T^{5} + 407351391 p^{3} T^{6} + 2536287349 p^{5} T^{7} + 40072406573 p^{6} T^{8} + 29313781287 p^{11} T^{9} + 423312129859 p^{12} T^{10} + 577745167595 p^{16} T^{11} + 62134032947693 p^{14} T^{12} + 157872405301449 p^{17} T^{13} + 62134032947693 p^{23} T^{14} + 577745167595 p^{34} T^{15} + 423312129859 p^{39} T^{16} + 29313781287 p^{47} T^{17} + 40072406573 p^{51} T^{18} + 2536287349 p^{59} T^{19} + 407351391 p^{66} T^{20} + 11044843 p^{75} T^{21} + 1665675 p^{82} T^{22} + 34807 p^{91} T^{23} + 2475 p^{99} T^{24} + p^{113} T^{25} + p^{117} T^{26} \) |
| 3 | \( 1 + 251 T + 13876 p^{2} T^{2} + 27806216 T^{3} + 8329657660 T^{4} + 546722041193 p T^{5} + 41931176049148 p^{2} T^{6} + 2451594657347855 p^{3} T^{7} + 159347881209274805 p^{4} T^{8} + 8421462319251896845 p^{5} T^{9} + \)\(48\!\cdots\!91\)\( p^{6} T^{10} + \)\(78\!\cdots\!54\)\( p^{8} T^{11} + \)\(15\!\cdots\!48\)\( p^{12} T^{12} + \)\(55\!\cdots\!12\)\( p^{9} T^{13} + \)\(15\!\cdots\!48\)\( p^{21} T^{14} + \)\(78\!\cdots\!54\)\( p^{26} T^{15} + \)\(48\!\cdots\!91\)\( p^{33} T^{16} + 8421462319251896845 p^{41} T^{17} + 159347881209274805 p^{49} T^{18} + 2451594657347855 p^{57} T^{19} + 41931176049148 p^{65} T^{20} + 546722041193 p^{73} T^{21} + 8329657660 p^{81} T^{22} + 27806216 p^{90} T^{23} + 13876 p^{101} T^{24} + 251 p^{108} T^{25} + p^{117} T^{26} \) | |
| 5 | \( 1 + 2159 T + 3188021 p T^{2} + 28160741197 T^{3} + 23935179306626 p T^{4} + 178415991081093909 T^{5} + \)\(57\!\cdots\!61\)\( T^{6} + \)\(14\!\cdots\!43\)\( p T^{7} + \)\(20\!\cdots\!28\)\( T^{8} + \)\(45\!\cdots\!69\)\( p T^{9} + \)\(22\!\cdots\!46\)\( p^{2} T^{10} + \)\(44\!\cdots\!56\)\( p^{3} T^{11} + \)\(20\!\cdots\!81\)\( p^{4} T^{12} + \)\(37\!\cdots\!78\)\( p^{5} T^{13} + \)\(20\!\cdots\!81\)\( p^{13} T^{14} + \)\(44\!\cdots\!56\)\( p^{21} T^{15} + \)\(22\!\cdots\!46\)\( p^{29} T^{16} + \)\(45\!\cdots\!69\)\( p^{37} T^{17} + \)\(20\!\cdots\!28\)\( p^{45} T^{18} + \)\(14\!\cdots\!43\)\( p^{55} T^{19} + \)\(57\!\cdots\!61\)\( p^{63} T^{20} + 178415991081093909 p^{72} T^{21} + 23935179306626 p^{82} T^{22} + 28160741197 p^{90} T^{23} + 3188021 p^{100} T^{24} + 2159 p^{108} T^{25} + p^{117} T^{26} \) | |
| 7 | \( 1 + 12576 T + 234740334 T^{2} + 2654952541846 T^{3} + 32555310154074404 T^{4} + 44456327745815940256 p T^{5} + \)\(44\!\cdots\!79\)\( p T^{6} + \)\(53\!\cdots\!20\)\( p^{2} T^{7} + \)\(67\!\cdots\!55\)\( p^{3} T^{8} + \)\(72\!\cdots\!04\)\( p^{4} T^{9} + \)\(80\!\cdots\!08\)\( p^{5} T^{10} + \)\(79\!\cdots\!94\)\( p^{6} T^{11} + \)\(11\!\cdots\!05\)\( p^{8} T^{12} + \)\(72\!\cdots\!64\)\( p^{8} T^{13} + \)\(11\!\cdots\!05\)\( p^{17} T^{14} + \)\(79\!\cdots\!94\)\( p^{24} T^{15} + \)\(80\!\cdots\!08\)\( p^{32} T^{16} + \)\(72\!\cdots\!04\)\( p^{40} T^{17} + \)\(67\!\cdots\!55\)\( p^{48} T^{18} + \)\(53\!\cdots\!20\)\( p^{56} T^{19} + \)\(44\!\cdots\!79\)\( p^{64} T^{20} + 44456327745815940256 p^{73} T^{21} + 32555310154074404 p^{81} T^{22} + 2654952541846 p^{90} T^{23} + 234740334 p^{99} T^{24} + 12576 p^{108} T^{25} + p^{117} T^{26} \) | |
| 11 | \( 1 + 112451 T + 18692441930 T^{2} + 1426364757461880 T^{3} + \)\(14\!\cdots\!66\)\( T^{4} + \)\(87\!\cdots\!75\)\( T^{5} + \)\(64\!\cdots\!96\)\( p T^{6} + \)\(36\!\cdots\!89\)\( T^{7} + \)\(26\!\cdots\!79\)\( T^{8} + \)\(12\!\cdots\!75\)\( T^{9} + \)\(83\!\cdots\!39\)\( T^{10} + \)\(37\!\cdots\!10\)\( T^{11} + \)\(23\!\cdots\!72\)\( T^{12} + \)\(95\!\cdots\!36\)\( T^{13} + \)\(23\!\cdots\!72\)\( p^{9} T^{14} + \)\(37\!\cdots\!10\)\( p^{18} T^{15} + \)\(83\!\cdots\!39\)\( p^{27} T^{16} + \)\(12\!\cdots\!75\)\( p^{36} T^{17} + \)\(26\!\cdots\!79\)\( p^{45} T^{18} + \)\(36\!\cdots\!89\)\( p^{54} T^{19} + \)\(64\!\cdots\!96\)\( p^{64} T^{20} + \)\(87\!\cdots\!75\)\( p^{72} T^{21} + \)\(14\!\cdots\!66\)\( p^{81} T^{22} + 1426364757461880 p^{90} T^{23} + 18692441930 p^{99} T^{24} + 112451 p^{108} T^{25} + p^{117} T^{26} \) | |
| 13 | \( 1 - 7129 T + 53225587737 T^{2} + 475313059409725 T^{3} + \)\(14\!\cdots\!02\)\( T^{4} + \)\(28\!\cdots\!65\)\( T^{5} + \)\(27\!\cdots\!13\)\( T^{6} + \)\(72\!\cdots\!71\)\( T^{7} + \)\(43\!\cdots\!32\)\( T^{8} + \)\(13\!\cdots\!29\)\( T^{9} + \)\(57\!\cdots\!90\)\( T^{10} + \)\(18\!\cdots\!60\)\( T^{11} + \)\(68\!\cdots\!33\)\( T^{12} + \)\(20\!\cdots\!70\)\( T^{13} + \)\(68\!\cdots\!33\)\( p^{9} T^{14} + \)\(18\!\cdots\!60\)\( p^{18} T^{15} + \)\(57\!\cdots\!90\)\( p^{27} T^{16} + \)\(13\!\cdots\!29\)\( p^{36} T^{17} + \)\(43\!\cdots\!32\)\( p^{45} T^{18} + \)\(72\!\cdots\!71\)\( p^{54} T^{19} + \)\(27\!\cdots\!13\)\( p^{63} T^{20} + \)\(28\!\cdots\!65\)\( p^{72} T^{21} + \)\(14\!\cdots\!02\)\( p^{81} T^{22} + 475313059409725 p^{90} T^{23} + 53225587737 p^{99} T^{24} - 7129 p^{108} T^{25} + p^{117} T^{26} \) | |
| 17 | \( 1 + 890862 T + 835612115137 T^{2} + 450271742899046400 T^{3} + \)\(25\!\cdots\!66\)\( T^{4} + \)\(60\!\cdots\!52\)\( p T^{5} + \)\(42\!\cdots\!82\)\( T^{6} + \)\(41\!\cdots\!36\)\( p^{2} T^{7} + \)\(31\!\cdots\!35\)\( T^{8} + \)\(14\!\cdots\!38\)\( T^{9} - \)\(25\!\cdots\!69\)\( T^{10} - \)\(21\!\cdots\!40\)\( T^{11} - \)\(98\!\cdots\!64\)\( T^{12} - \)\(37\!\cdots\!64\)\( T^{13} - \)\(98\!\cdots\!64\)\( p^{9} T^{14} - \)\(21\!\cdots\!40\)\( p^{18} T^{15} - \)\(25\!\cdots\!69\)\( p^{27} T^{16} + \)\(14\!\cdots\!38\)\( p^{36} T^{17} + \)\(31\!\cdots\!35\)\( p^{45} T^{18} + \)\(41\!\cdots\!36\)\( p^{56} T^{19} + \)\(42\!\cdots\!82\)\( p^{63} T^{20} + \)\(60\!\cdots\!52\)\( p^{73} T^{21} + \)\(25\!\cdots\!66\)\( p^{81} T^{22} + 450271742899046400 p^{90} T^{23} + 835612115137 p^{99} T^{24} + 890862 p^{108} T^{25} + p^{117} T^{26} \) | |
| 19 | \( 1 + 1435874 T + 3563337615835 T^{2} + 3990865754595402512 T^{3} + \)\(57\!\cdots\!74\)\( T^{4} + \)\(52\!\cdots\!88\)\( T^{5} + \)\(55\!\cdots\!90\)\( T^{6} + \)\(43\!\cdots\!88\)\( T^{7} + \)\(38\!\cdots\!67\)\( T^{8} + \)\(25\!\cdots\!26\)\( T^{9} + \)\(19\!\cdots\!09\)\( T^{10} + \)\(11\!\cdots\!28\)\( T^{11} + \)\(77\!\cdots\!64\)\( T^{12} + \)\(41\!\cdots\!68\)\( T^{13} + \)\(77\!\cdots\!64\)\( p^{9} T^{14} + \)\(11\!\cdots\!28\)\( p^{18} T^{15} + \)\(19\!\cdots\!09\)\( p^{27} T^{16} + \)\(25\!\cdots\!26\)\( p^{36} T^{17} + \)\(38\!\cdots\!67\)\( p^{45} T^{18} + \)\(43\!\cdots\!88\)\( p^{54} T^{19} + \)\(55\!\cdots\!90\)\( p^{63} T^{20} + \)\(52\!\cdots\!88\)\( p^{72} T^{21} + \)\(57\!\cdots\!74\)\( p^{81} T^{22} + 3990865754595402512 p^{90} T^{23} + 3563337615835 p^{99} T^{24} + 1435874 p^{108} T^{25} + p^{117} T^{26} \) | |
| 23 | \( 1 + 2565799 T + 9930515111755 T^{2} + 9921467271054874757 T^{3} + \)\(19\!\cdots\!94\)\( T^{4} - \)\(15\!\cdots\!51\)\( T^{5} + \)\(18\!\cdots\!25\)\( T^{6} - \)\(47\!\cdots\!69\)\( T^{7} + \)\(12\!\cdots\!02\)\( T^{8} + \)\(13\!\cdots\!89\)\( T^{9} + \)\(41\!\cdots\!44\)\( T^{10} + \)\(72\!\cdots\!24\)\( T^{11} + \)\(11\!\cdots\!19\)\( T^{12} - \)\(61\!\cdots\!86\)\( T^{13} + \)\(11\!\cdots\!19\)\( p^{9} T^{14} + \)\(72\!\cdots\!24\)\( p^{18} T^{15} + \)\(41\!\cdots\!44\)\( p^{27} T^{16} + \)\(13\!\cdots\!89\)\( p^{36} T^{17} + \)\(12\!\cdots\!02\)\( p^{45} T^{18} - \)\(47\!\cdots\!69\)\( p^{54} T^{19} + \)\(18\!\cdots\!25\)\( p^{63} T^{20} - \)\(15\!\cdots\!51\)\( p^{72} T^{21} + \)\(19\!\cdots\!94\)\( p^{81} T^{22} + 9921467271054874757 p^{90} T^{23} + 9930515111755 p^{99} T^{24} + 2565799 p^{108} T^{25} + p^{117} T^{26} \) | |
| 29 | \( 1 + 2992323 T + 127184618851209 T^{2} + 11848689807404482629 p T^{3} + \)\(77\!\cdots\!98\)\( T^{4} + \)\(19\!\cdots\!13\)\( T^{5} + \)\(30\!\cdots\!65\)\( T^{6} + \)\(75\!\cdots\!15\)\( T^{7} + \)\(88\!\cdots\!40\)\( T^{8} + \)\(21\!\cdots\!65\)\( T^{9} + \)\(19\!\cdots\!62\)\( T^{10} + \)\(44\!\cdots\!32\)\( T^{11} + \)\(12\!\cdots\!17\)\( p T^{12} + \)\(73\!\cdots\!18\)\( T^{13} + \)\(12\!\cdots\!17\)\( p^{10} T^{14} + \)\(44\!\cdots\!32\)\( p^{18} T^{15} + \)\(19\!\cdots\!62\)\( p^{27} T^{16} + \)\(21\!\cdots\!65\)\( p^{36} T^{17} + \)\(88\!\cdots\!40\)\( p^{45} T^{18} + \)\(75\!\cdots\!15\)\( p^{54} T^{19} + \)\(30\!\cdots\!65\)\( p^{63} T^{20} + \)\(19\!\cdots\!13\)\( p^{72} T^{21} + \)\(77\!\cdots\!98\)\( p^{81} T^{22} + 11848689807404482629 p^{91} T^{23} + 127184618851209 p^{99} T^{24} + 2992323 p^{108} T^{25} + p^{117} T^{26} \) | |
| 31 | \( 1 + 8242245 T + 172905465094401 T^{2} + \)\(11\!\cdots\!55\)\( T^{3} + \)\(14\!\cdots\!54\)\( T^{4} + \)\(85\!\cdots\!75\)\( T^{5} + \)\(26\!\cdots\!05\)\( p T^{6} + \)\(43\!\cdots\!53\)\( T^{7} + \)\(36\!\cdots\!90\)\( T^{8} + \)\(17\!\cdots\!35\)\( T^{9} + \)\(13\!\cdots\!78\)\( T^{10} + \)\(60\!\cdots\!28\)\( T^{11} + \)\(41\!\cdots\!57\)\( T^{12} + \)\(17\!\cdots\!22\)\( T^{13} + \)\(41\!\cdots\!57\)\( p^{9} T^{14} + \)\(60\!\cdots\!28\)\( p^{18} T^{15} + \)\(13\!\cdots\!78\)\( p^{27} T^{16} + \)\(17\!\cdots\!35\)\( p^{36} T^{17} + \)\(36\!\cdots\!90\)\( p^{45} T^{18} + \)\(43\!\cdots\!53\)\( p^{54} T^{19} + \)\(26\!\cdots\!05\)\( p^{64} T^{20} + \)\(85\!\cdots\!75\)\( p^{72} T^{21} + \)\(14\!\cdots\!54\)\( p^{81} T^{22} + \)\(11\!\cdots\!55\)\( p^{90} T^{23} + 172905465094401 p^{99} T^{24} + 8242245 p^{108} T^{25} + p^{117} T^{26} \) | |
| 41 | \( 1 + 50975109 T + 3707325767667056 T^{2} + \)\(15\!\cdots\!94\)\( T^{3} + \)\(64\!\cdots\!44\)\( T^{4} + \)\(21\!\cdots\!81\)\( T^{5} + \)\(70\!\cdots\!18\)\( T^{6} + \)\(19\!\cdots\!35\)\( T^{7} + \)\(53\!\cdots\!85\)\( T^{8} + \)\(12\!\cdots\!87\)\( T^{9} + \)\(29\!\cdots\!51\)\( T^{10} + \)\(63\!\cdots\!72\)\( T^{11} + \)\(12\!\cdots\!38\)\( T^{12} + \)\(57\!\cdots\!38\)\( p T^{13} + \)\(12\!\cdots\!38\)\( p^{9} T^{14} + \)\(63\!\cdots\!72\)\( p^{18} T^{15} + \)\(29\!\cdots\!51\)\( p^{27} T^{16} + \)\(12\!\cdots\!87\)\( p^{36} T^{17} + \)\(53\!\cdots\!85\)\( p^{45} T^{18} + \)\(19\!\cdots\!35\)\( p^{54} T^{19} + \)\(70\!\cdots\!18\)\( p^{63} T^{20} + \)\(21\!\cdots\!81\)\( p^{72} T^{21} + \)\(64\!\cdots\!44\)\( p^{81} T^{22} + \)\(15\!\cdots\!94\)\( p^{90} T^{23} + 3707325767667056 p^{99} T^{24} + 50975109 p^{108} T^{25} + p^{117} T^{26} \) | |
| 43 | \( 1 + 18142836 T + 4379055664284607 T^{2} + \)\(92\!\cdots\!84\)\( T^{3} + \)\(95\!\cdots\!30\)\( T^{4} + \)\(21\!\cdots\!44\)\( T^{5} + \)\(13\!\cdots\!78\)\( T^{6} + \)\(30\!\cdots\!44\)\( T^{7} + \)\(14\!\cdots\!55\)\( T^{8} + \)\(30\!\cdots\!92\)\( T^{9} + \)\(11\!\cdots\!41\)\( T^{10} + \)\(23\!\cdots\!76\)\( T^{11} + \)\(73\!\cdots\!64\)\( T^{12} + \)\(13\!\cdots\!64\)\( T^{13} + \)\(73\!\cdots\!64\)\( p^{9} T^{14} + \)\(23\!\cdots\!76\)\( p^{18} T^{15} + \)\(11\!\cdots\!41\)\( p^{27} T^{16} + \)\(30\!\cdots\!92\)\( p^{36} T^{17} + \)\(14\!\cdots\!55\)\( p^{45} T^{18} + \)\(30\!\cdots\!44\)\( p^{54} T^{19} + \)\(13\!\cdots\!78\)\( p^{63} T^{20} + \)\(21\!\cdots\!44\)\( p^{72} T^{21} + \)\(95\!\cdots\!30\)\( p^{81} T^{22} + \)\(92\!\cdots\!84\)\( p^{90} T^{23} + 4379055664284607 p^{99} T^{24} + 18142836 p^{108} T^{25} + p^{117} T^{26} \) | |
| 47 | \( 1 + 14353596 T + 6654867111737854 T^{2} + \)\(79\!\cdots\!70\)\( T^{3} + \)\(23\!\cdots\!68\)\( T^{4} + \)\(22\!\cdots\!60\)\( T^{5} + \)\(58\!\cdots\!85\)\( T^{6} + \)\(40\!\cdots\!72\)\( T^{7} + \)\(11\!\cdots\!61\)\( T^{8} + \)\(53\!\cdots\!36\)\( T^{9} + \)\(17\!\cdots\!40\)\( T^{10} + \)\(56\!\cdots\!38\)\( T^{11} + \)\(22\!\cdots\!41\)\( T^{12} + \)\(59\!\cdots\!48\)\( T^{13} + \)\(22\!\cdots\!41\)\( p^{9} T^{14} + \)\(56\!\cdots\!38\)\( p^{18} T^{15} + \)\(17\!\cdots\!40\)\( p^{27} T^{16} + \)\(53\!\cdots\!36\)\( p^{36} T^{17} + \)\(11\!\cdots\!61\)\( p^{45} T^{18} + \)\(40\!\cdots\!72\)\( p^{54} T^{19} + \)\(58\!\cdots\!85\)\( p^{63} T^{20} + \)\(22\!\cdots\!60\)\( p^{72} T^{21} + \)\(23\!\cdots\!68\)\( p^{81} T^{22} + \)\(79\!\cdots\!70\)\( p^{90} T^{23} + 6654867111737854 p^{99} T^{24} + 14353596 p^{108} T^{25} + p^{117} T^{26} \) | |
| 53 | \( 1 - 74438872 T + 25489256101766872 T^{2} - \)\(14\!\cdots\!38\)\( T^{3} + \)\(29\!\cdots\!90\)\( T^{4} - \)\(13\!\cdots\!48\)\( T^{5} + \)\(21\!\cdots\!11\)\( T^{6} - \)\(78\!\cdots\!36\)\( T^{7} + \)\(11\!\cdots\!21\)\( T^{8} - \)\(31\!\cdots\!12\)\( T^{9} + \)\(45\!\cdots\!86\)\( T^{10} - \)\(95\!\cdots\!74\)\( T^{11} + \)\(15\!\cdots\!95\)\( T^{12} - \)\(29\!\cdots\!76\)\( T^{13} + \)\(15\!\cdots\!95\)\( p^{9} T^{14} - \)\(95\!\cdots\!74\)\( p^{18} T^{15} + \)\(45\!\cdots\!86\)\( p^{27} T^{16} - \)\(31\!\cdots\!12\)\( p^{36} T^{17} + \)\(11\!\cdots\!21\)\( p^{45} T^{18} - \)\(78\!\cdots\!36\)\( p^{54} T^{19} + \)\(21\!\cdots\!11\)\( p^{63} T^{20} - \)\(13\!\cdots\!48\)\( p^{72} T^{21} + \)\(29\!\cdots\!90\)\( p^{81} T^{22} - \)\(14\!\cdots\!38\)\( p^{90} T^{23} + 25489256101766872 p^{99} T^{24} - 74438872 p^{108} T^{25} + p^{117} T^{26} \) | |
| 59 | \( 1 + 251964328 T + 75393099916427851 T^{2} + \)\(13\!\cdots\!00\)\( T^{3} + \)\(25\!\cdots\!46\)\( T^{4} + \)\(36\!\cdots\!96\)\( T^{5} + \)\(54\!\cdots\!50\)\( T^{6} + \)\(68\!\cdots\!92\)\( T^{7} + \)\(88\!\cdots\!27\)\( T^{8} + \)\(99\!\cdots\!40\)\( T^{9} + \)\(11\!\cdots\!61\)\( T^{10} + \)\(11\!\cdots\!20\)\( T^{11} + \)\(11\!\cdots\!60\)\( T^{12} + \)\(11\!\cdots\!88\)\( T^{13} + \)\(11\!\cdots\!60\)\( p^{9} T^{14} + \)\(11\!\cdots\!20\)\( p^{18} T^{15} + \)\(11\!\cdots\!61\)\( p^{27} T^{16} + \)\(99\!\cdots\!40\)\( p^{36} T^{17} + \)\(88\!\cdots\!27\)\( p^{45} T^{18} + \)\(68\!\cdots\!92\)\( p^{54} T^{19} + \)\(54\!\cdots\!50\)\( p^{63} T^{20} + \)\(36\!\cdots\!96\)\( p^{72} T^{21} + \)\(25\!\cdots\!46\)\( p^{81} T^{22} + \)\(13\!\cdots\!00\)\( p^{90} T^{23} + 75393099916427851 p^{99} T^{24} + 251964328 p^{108} T^{25} + p^{117} T^{26} \) | |
| 61 | \( 1 - 202847323 T + 89929011032389033 T^{2} - \)\(16\!\cdots\!77\)\( T^{3} + \)\(43\!\cdots\!62\)\( T^{4} - \)\(72\!\cdots\!13\)\( T^{5} + \)\(14\!\cdots\!29\)\( T^{6} - \)\(21\!\cdots\!19\)\( T^{7} + \)\(54\!\cdots\!72\)\( p T^{8} - \)\(44\!\cdots\!69\)\( T^{9} + \)\(61\!\cdots\!78\)\( T^{10} - \)\(74\!\cdots\!72\)\( T^{11} + \)\(89\!\cdots\!53\)\( T^{12} - \)\(97\!\cdots\!50\)\( T^{13} + \)\(89\!\cdots\!53\)\( p^{9} T^{14} - \)\(74\!\cdots\!72\)\( p^{18} T^{15} + \)\(61\!\cdots\!78\)\( p^{27} T^{16} - \)\(44\!\cdots\!69\)\( p^{36} T^{17} + \)\(54\!\cdots\!72\)\( p^{46} T^{18} - \)\(21\!\cdots\!19\)\( p^{54} T^{19} + \)\(14\!\cdots\!29\)\( p^{63} T^{20} - \)\(72\!\cdots\!13\)\( p^{72} T^{21} + \)\(43\!\cdots\!62\)\( p^{81} T^{22} - \)\(16\!\cdots\!77\)\( p^{90} T^{23} + 89929011032389033 p^{99} T^{24} - 202847323 p^{108} T^{25} + p^{117} T^{26} \) | |
| 67 | \( 1 + 12257509 T + 167264900892509273 T^{2} - \)\(46\!\cdots\!91\)\( T^{3} + \)\(14\!\cdots\!58\)\( T^{4} - \)\(74\!\cdots\!43\)\( T^{5} + \)\(13\!\cdots\!97\)\( p T^{6} - \)\(55\!\cdots\!87\)\( T^{7} + \)\(41\!\cdots\!38\)\( T^{8} - \)\(27\!\cdots\!09\)\( T^{9} + \)\(16\!\cdots\!18\)\( T^{10} - \)\(10\!\cdots\!30\)\( T^{11} + \)\(52\!\cdots\!41\)\( T^{12} - \)\(31\!\cdots\!18\)\( T^{13} + \)\(52\!\cdots\!41\)\( p^{9} T^{14} - \)\(10\!\cdots\!30\)\( p^{18} T^{15} + \)\(16\!\cdots\!18\)\( p^{27} T^{16} - \)\(27\!\cdots\!09\)\( p^{36} T^{17} + \)\(41\!\cdots\!38\)\( p^{45} T^{18} - \)\(55\!\cdots\!87\)\( p^{54} T^{19} + \)\(13\!\cdots\!97\)\( p^{64} T^{20} - \)\(74\!\cdots\!43\)\( p^{72} T^{21} + \)\(14\!\cdots\!58\)\( p^{81} T^{22} - \)\(46\!\cdots\!91\)\( p^{90} T^{23} + 167264900892509273 p^{99} T^{24} + 12257509 p^{108} T^{25} + p^{117} T^{26} \) | |
| 71 | \( 1 + 310979094 T + 427559593428276902 T^{2} + \)\(11\!\cdots\!20\)\( T^{3} + \)\(82\!\cdots\!28\)\( T^{4} + \)\(18\!\cdots\!18\)\( T^{5} + \)\(94\!\cdots\!97\)\( T^{6} + \)\(18\!\cdots\!12\)\( T^{7} + \)\(72\!\cdots\!81\)\( T^{8} + \)\(11\!\cdots\!02\)\( T^{9} + \)\(40\!\cdots\!96\)\( T^{10} + \)\(57\!\cdots\!44\)\( T^{11} + \)\(19\!\cdots\!49\)\( T^{12} + \)\(25\!\cdots\!24\)\( T^{13} + \)\(19\!\cdots\!49\)\( p^{9} T^{14} + \)\(57\!\cdots\!44\)\( p^{18} T^{15} + \)\(40\!\cdots\!96\)\( p^{27} T^{16} + \)\(11\!\cdots\!02\)\( p^{36} T^{17} + \)\(72\!\cdots\!81\)\( p^{45} T^{18} + \)\(18\!\cdots\!12\)\( p^{54} T^{19} + \)\(94\!\cdots\!97\)\( p^{63} T^{20} + \)\(18\!\cdots\!18\)\( p^{72} T^{21} + \)\(82\!\cdots\!28\)\( p^{81} T^{22} + \)\(11\!\cdots\!20\)\( p^{90} T^{23} + 427559593428276902 p^{99} T^{24} + 310979094 p^{108} T^{25} + p^{117} T^{26} \) | |
| 73 | \( 1 + 249752015 T + 441837081096862684 T^{2} + \)\(76\!\cdots\!22\)\( T^{3} + \)\(89\!\cdots\!00\)\( T^{4} + \)\(10\!\cdots\!91\)\( T^{5} + \)\(11\!\cdots\!38\)\( T^{6} + \)\(10\!\cdots\!33\)\( T^{7} + \)\(16\!\cdots\!89\)\( p T^{8} + \)\(80\!\cdots\!49\)\( T^{9} + \)\(97\!\cdots\!99\)\( T^{10} + \)\(51\!\cdots\!48\)\( T^{11} + \)\(66\!\cdots\!70\)\( T^{12} + \)\(30\!\cdots\!10\)\( T^{13} + \)\(66\!\cdots\!70\)\( p^{9} T^{14} + \)\(51\!\cdots\!48\)\( p^{18} T^{15} + \)\(97\!\cdots\!99\)\( p^{27} T^{16} + \)\(80\!\cdots\!49\)\( p^{36} T^{17} + \)\(16\!\cdots\!89\)\( p^{46} T^{18} + \)\(10\!\cdots\!33\)\( p^{54} T^{19} + \)\(11\!\cdots\!38\)\( p^{63} T^{20} + \)\(10\!\cdots\!91\)\( p^{72} T^{21} + \)\(89\!\cdots\!00\)\( p^{81} T^{22} + \)\(76\!\cdots\!22\)\( p^{90} T^{23} + 441837081096862684 p^{99} T^{24} + 249752015 p^{108} T^{25} + p^{117} T^{26} \) | |
| 79 | \( 1 - 30429049 T + 933972853246714827 T^{2} - \)\(13\!\cdots\!91\)\( T^{3} + \)\(42\!\cdots\!34\)\( T^{4} + \)\(11\!\cdots\!25\)\( T^{5} + \)\(12\!\cdots\!57\)\( T^{6} + \)\(66\!\cdots\!71\)\( T^{7} + \)\(27\!\cdots\!46\)\( T^{8} + \)\(19\!\cdots\!85\)\( T^{9} + \)\(48\!\cdots\!00\)\( T^{10} + \)\(39\!\cdots\!84\)\( T^{11} + \)\(69\!\cdots\!31\)\( T^{12} + \)\(55\!\cdots\!34\)\( T^{13} + \)\(69\!\cdots\!31\)\( p^{9} T^{14} + \)\(39\!\cdots\!84\)\( p^{18} T^{15} + \)\(48\!\cdots\!00\)\( p^{27} T^{16} + \)\(19\!\cdots\!85\)\( p^{36} T^{17} + \)\(27\!\cdots\!46\)\( p^{45} T^{18} + \)\(66\!\cdots\!71\)\( p^{54} T^{19} + \)\(12\!\cdots\!57\)\( p^{63} T^{20} + \)\(11\!\cdots\!25\)\( p^{72} T^{21} + \)\(42\!\cdots\!34\)\( p^{81} T^{22} - \)\(13\!\cdots\!91\)\( p^{90} T^{23} + 933972853246714827 p^{99} T^{24} - 30429049 p^{108} T^{25} + p^{117} T^{26} \) | |
| 83 | \( 1 + 2559788658 T + 4217259111510254214 T^{2} + \)\(52\!\cdots\!84\)\( T^{3} + \)\(53\!\cdots\!08\)\( T^{4} + \)\(46\!\cdots\!94\)\( T^{5} + \)\(35\!\cdots\!65\)\( T^{6} + \)\(24\!\cdots\!72\)\( T^{7} + \)\(15\!\cdots\!21\)\( T^{8} + \)\(91\!\cdots\!02\)\( T^{9} + \)\(49\!\cdots\!20\)\( T^{10} + \)\(25\!\cdots\!80\)\( T^{11} + \)\(11\!\cdots\!13\)\( T^{12} + \)\(53\!\cdots\!64\)\( T^{13} + \)\(11\!\cdots\!13\)\( p^{9} T^{14} + \)\(25\!\cdots\!80\)\( p^{18} T^{15} + \)\(49\!\cdots\!20\)\( p^{27} T^{16} + \)\(91\!\cdots\!02\)\( p^{36} T^{17} + \)\(15\!\cdots\!21\)\( p^{45} T^{18} + \)\(24\!\cdots\!72\)\( p^{54} T^{19} + \)\(35\!\cdots\!65\)\( p^{63} T^{20} + \)\(46\!\cdots\!94\)\( p^{72} T^{21} + \)\(53\!\cdots\!08\)\( p^{81} T^{22} + \)\(52\!\cdots\!84\)\( p^{90} T^{23} + 4217259111510254214 p^{99} T^{24} + 2559788658 p^{108} T^{25} + p^{117} T^{26} \) | |
| 89 | \( 1 + 3063565514 T + 7115585762582358237 T^{2} + \)\(11\!\cdots\!48\)\( T^{3} + \)\(16\!\cdots\!78\)\( T^{4} + \)\(19\!\cdots\!00\)\( T^{5} + \)\(20\!\cdots\!18\)\( T^{6} + \)\(18\!\cdots\!72\)\( T^{7} + \)\(16\!\cdots\!15\)\( T^{8} + \)\(12\!\cdots\!46\)\( T^{9} + \)\(92\!\cdots\!39\)\( T^{10} + \)\(63\!\cdots\!76\)\( T^{11} + \)\(40\!\cdots\!80\)\( T^{12} + \)\(24\!\cdots\!32\)\( T^{13} + \)\(40\!\cdots\!80\)\( p^{9} T^{14} + \)\(63\!\cdots\!76\)\( p^{18} T^{15} + \)\(92\!\cdots\!39\)\( p^{27} T^{16} + \)\(12\!\cdots\!46\)\( p^{36} T^{17} + \)\(16\!\cdots\!15\)\( p^{45} T^{18} + \)\(18\!\cdots\!72\)\( p^{54} T^{19} + \)\(20\!\cdots\!18\)\( p^{63} T^{20} + \)\(19\!\cdots\!00\)\( p^{72} T^{21} + \)\(16\!\cdots\!78\)\( p^{81} T^{22} + \)\(11\!\cdots\!48\)\( p^{90} T^{23} + 7115585762582358237 p^{99} T^{24} + 3063565514 p^{108} T^{25} + p^{117} T^{26} \) | |
| 97 | \( 1 + 429307758 T + 5018988333288425617 T^{2} + \)\(19\!\cdots\!84\)\( T^{3} + \)\(12\!\cdots\!26\)\( T^{4} + \)\(41\!\cdots\!40\)\( T^{5} + \)\(20\!\cdots\!78\)\( T^{6} + \)\(57\!\cdots\!00\)\( T^{7} + \)\(26\!\cdots\!91\)\( T^{8} + \)\(58\!\cdots\!18\)\( T^{9} + \)\(26\!\cdots\!75\)\( T^{10} + \)\(48\!\cdots\!96\)\( T^{11} + \)\(23\!\cdots\!84\)\( T^{12} + \)\(36\!\cdots\!52\)\( T^{13} + \)\(23\!\cdots\!84\)\( p^{9} T^{14} + \)\(48\!\cdots\!96\)\( p^{18} T^{15} + \)\(26\!\cdots\!75\)\( p^{27} T^{16} + \)\(58\!\cdots\!18\)\( p^{36} T^{17} + \)\(26\!\cdots\!91\)\( p^{45} T^{18} + \)\(57\!\cdots\!00\)\( p^{54} T^{19} + \)\(20\!\cdots\!78\)\( p^{63} T^{20} + \)\(41\!\cdots\!40\)\( p^{72} T^{21} + \)\(12\!\cdots\!26\)\( p^{81} T^{22} + \)\(19\!\cdots\!84\)\( p^{90} T^{23} + 5018988333288425617 p^{99} T^{24} + 429307758 p^{108} T^{25} + p^{117} T^{26} \) | |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{26} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−4.70010643160135096236676517739, −4.35165554778423436660424512217, −4.34826229053114272615957026082, −4.22732028993501521076732691672, −4.12376238870659213449523118922, −3.91822948828322876533145225694, −3.89890449603083150680251727105, −3.79317454522466586950446214323, −3.67176701087675617548585023748, −3.63911127656554038524547518384, −3.43693428072588755204388600184, −3.11281488573955380179819673017, −2.85414154134567379323323563665, −2.79892032887819332055712135616, −2.74782461753150060881516511978, −2.55102539853698195198170479651, −2.44238278700291967148090768601, −2.19994020544372368815017882173, −2.16953662266911632620252124714, −1.85189746689486432281318875799, −1.56301998421231030290602563923, −1.51859688804889896267310212480, −1.51829705786988136337311170308, −1.14740040801441444243765740808, −1.04211091021038006701599095220, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1.04211091021038006701599095220, 1.14740040801441444243765740808, 1.51829705786988136337311170308, 1.51859688804889896267310212480, 1.56301998421231030290602563923, 1.85189746689486432281318875799, 2.16953662266911632620252124714, 2.19994020544372368815017882173, 2.44238278700291967148090768601, 2.55102539853698195198170479651, 2.74782461753150060881516511978, 2.79892032887819332055712135616, 2.85414154134567379323323563665, 3.11281488573955380179819673017, 3.43693428072588755204388600184, 3.63911127656554038524547518384, 3.67176701087675617548585023748, 3.79317454522466586950446214323, 3.89890449603083150680251727105, 3.91822948828322876533145225694, 4.12376238870659213449523118922, 4.22732028993501521076732691672, 4.34826229053114272615957026082, 4.35165554778423436660424512217, 4.70010643160135096236676517739
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.