Dirichlet series
| L(s) = 1 | − 2·2-s − 4·3-s + 2·4-s + 166·5-s + 8·6-s + 5.30e3·7-s + 3.31e3·8-s − 3.93e4·9-s − 332·10-s − 3.15e4·11-s − 8·12-s + 7.13e4·13-s − 1.06e4·14-s − 664·15-s + 2.64e4·16-s + 7.87e4·18-s + 1.00e5·19-s + 332·20-s − 2.12e4·21-s + 6.31e4·22-s − 1.32e4·24-s + 1.37e4·25-s − 1.42e5·26-s + 2.87e5·27-s + 1.06e4·28-s − 2.47e6·29-s + 1.32e3·30-s + ⋯ |
| L(s) = 1 | − 1/8·2-s − 0.0493·3-s + 0.00781·4-s + 0.265·5-s + 0.00617·6-s + 2.21·7-s + 0.809·8-s − 5.99·9-s − 0.0331·10-s − 2.15·11-s − 0.000385·12-s + 2.49·13-s − 0.276·14-s − 0.0131·15-s + 0.404·16-s + 0.749·18-s + 0.769·19-s + 0.00207·20-s − 0.109·21-s + 0.269·22-s − 0.0399·24-s + 0.0352·25-s − 0.312·26-s + 0.541·27-s + 0.0172·28-s − 3.50·29-s + 0.00163·30-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(13^{18}\right)^{s/2} \, \Gamma_{\C}(s)^{18} \, L(s)\cr=\mathstrut & \,\Lambda(9-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(13^{18}\right)^{s/2} \, \Gamma_{\C}(s+4)^{18} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(36\) |
| Conductor: | \(13^{18}\) |
| Sign: | $1$ |
| Analytic conductor: | \(1.07386\times 10^{13}\) |
| Root analytic conductor: | \(2.30128\) |
| Motivic weight: | \(8\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((36,\ 13^{18} ,\ ( \ : [4]^{18} ),\ 1 )\) |
Particular Values
| \(L(\frac{9}{2})\) | \(\approx\) | \(0.001687197481\) |
| \(L(\frac12)\) | \(\approx\) | \(0.001687197481\) |
| \(L(5)\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 13 | \( 1 - 71300 T + 195917019 p T^{2} - 412833402104 p^{2} T^{3} + 71122709780210 p^{4} T^{4} - 1330347199541744 p^{7} T^{5} + 4392822198320369664 p^{8} T^{6} - \)\(82\!\cdots\!88\)\( p^{10} T^{7} + \)\(95\!\cdots\!36\)\( p^{13} T^{8} - \)\(11\!\cdots\!56\)\( p^{16} T^{9} + \)\(95\!\cdots\!36\)\( p^{21} T^{10} - \)\(82\!\cdots\!88\)\( p^{26} T^{11} + 4392822198320369664 p^{32} T^{12} - 1330347199541744 p^{39} T^{13} + 71122709780210 p^{44} T^{14} - 412833402104 p^{50} T^{15} + 195917019 p^{57} T^{16} - 71300 p^{64} T^{17} + p^{72} T^{18} \) |
| good | 2 | \( 1 + p T + p T^{2} - 829 p^{2} T^{3} - 39759 T^{4} + 136715 p T^{5} + 3062153 p T^{6} + 1753271 p^{5} T^{7} + 69708507 p^{6} T^{8} - 147011969 p^{7} T^{9} + 377167339 p^{7} T^{10} - 8834064613 p^{12} T^{11} - 85235647069 p^{12} T^{12} + 820773548509 p^{13} T^{13} + 12836724942907 p^{13} T^{14} + 11452015992103 p^{17} T^{15} - 3132626508999 p^{18} T^{16} - 737758678421719 p^{19} T^{17} + 648091651617689 p^{19} T^{18} - 737758678421719 p^{27} T^{19} - 3132626508999 p^{34} T^{20} + 11452015992103 p^{41} T^{21} + 12836724942907 p^{45} T^{22} + 820773548509 p^{53} T^{23} - 85235647069 p^{60} T^{24} - 8834064613 p^{68} T^{25} + 377167339 p^{71} T^{26} - 147011969 p^{79} T^{27} + 69708507 p^{86} T^{28} + 1753271 p^{93} T^{29} + 3062153 p^{97} T^{30} + 136715 p^{105} T^{31} - 39759 p^{112} T^{32} - 829 p^{122} T^{33} + p^{129} T^{34} + p^{137} T^{35} + p^{144} T^{36} \) |
| 3 | \( ( 1 + 2 T + 6562 p T^{2} - 8608 p T^{3} + 21028954 p^{2} T^{4} - 391486270 p^{2} T^{5} + 16987864951 p^{4} T^{6} - 639749469916 p^{4} T^{7} + 11993654966464 p^{6} T^{8} - 61019208852064 p^{8} T^{9} + 11993654966464 p^{14} T^{10} - 639749469916 p^{20} T^{11} + 16987864951 p^{28} T^{12} - 391486270 p^{34} T^{13} + 21028954 p^{42} T^{14} - 8608 p^{49} T^{15} + 6562 p^{57} T^{16} + 2 p^{64} T^{17} + p^{72} T^{18} )^{2} \) | |
| 5 | \( 1 - 166 T + 13778 T^{2} + 660391826 T^{3} - 306716952156 T^{4} - 228485743029026 T^{5} + 260213261433430822 T^{6} - 42554293358413384954 p T^{7} - \)\(48\!\cdots\!16\)\( p^{2} T^{8} + \)\(10\!\cdots\!82\)\( p^{3} T^{9} - \)\(88\!\cdots\!34\)\( p^{4} T^{10} - \)\(82\!\cdots\!66\)\( p^{5} T^{11} + \)\(39\!\cdots\!91\)\( p^{6} T^{12} - \)\(16\!\cdots\!04\)\( p^{7} T^{13} - \)\(22\!\cdots\!84\)\( p^{8} T^{14} + \)\(84\!\cdots\!88\)\( p^{9} T^{15} - \)\(61\!\cdots\!76\)\( p^{10} T^{16} - \)\(87\!\cdots\!16\)\( p^{11} T^{17} + \)\(14\!\cdots\!72\)\( p^{12} T^{18} - \)\(87\!\cdots\!16\)\( p^{19} T^{19} - \)\(61\!\cdots\!76\)\( p^{26} T^{20} + \)\(84\!\cdots\!88\)\( p^{33} T^{21} - \)\(22\!\cdots\!84\)\( p^{40} T^{22} - \)\(16\!\cdots\!04\)\( p^{47} T^{23} + \)\(39\!\cdots\!91\)\( p^{54} T^{24} - \)\(82\!\cdots\!66\)\( p^{61} T^{25} - \)\(88\!\cdots\!34\)\( p^{68} T^{26} + \)\(10\!\cdots\!82\)\( p^{75} T^{27} - \)\(48\!\cdots\!16\)\( p^{82} T^{28} - 42554293358413384954 p^{89} T^{29} + 260213261433430822 p^{96} T^{30} - 228485743029026 p^{104} T^{31} - 306716952156 p^{112} T^{32} + 660391826 p^{120} T^{33} + 13778 p^{128} T^{34} - 166 p^{136} T^{35} + p^{144} T^{36} \) | |
| 7 | \( 1 - 5308 T + 14087432 T^{2} - 22301615936 T^{3} - 44226969761316 T^{4} + 34841884131333708 p T^{5} - 8629787367894189712 p^{2} T^{6} + \)\(72\!\cdots\!68\)\( p^{3} T^{7} + \)\(11\!\cdots\!60\)\( p^{4} T^{8} - \)\(97\!\cdots\!36\)\( p^{6} T^{9} + \)\(22\!\cdots\!12\)\( p^{6} T^{10} - \)\(66\!\cdots\!92\)\( p^{7} T^{11} + \)\(21\!\cdots\!31\)\( p^{8} T^{12} + \)\(66\!\cdots\!48\)\( p^{9} T^{13} - \)\(27\!\cdots\!28\)\( p^{10} T^{14} + \)\(79\!\cdots\!96\)\( p^{11} T^{15} + \)\(74\!\cdots\!24\)\( p^{12} T^{16} - \)\(14\!\cdots\!40\)\( p^{13} T^{17} + \)\(54\!\cdots\!20\)\( p^{14} T^{18} - \)\(14\!\cdots\!40\)\( p^{21} T^{19} + \)\(74\!\cdots\!24\)\( p^{28} T^{20} + \)\(79\!\cdots\!96\)\( p^{35} T^{21} - \)\(27\!\cdots\!28\)\( p^{42} T^{22} + \)\(66\!\cdots\!48\)\( p^{49} T^{23} + \)\(21\!\cdots\!31\)\( p^{56} T^{24} - \)\(66\!\cdots\!92\)\( p^{63} T^{25} + \)\(22\!\cdots\!12\)\( p^{70} T^{26} - \)\(97\!\cdots\!36\)\( p^{78} T^{27} + \)\(11\!\cdots\!60\)\( p^{84} T^{28} + \)\(72\!\cdots\!68\)\( p^{91} T^{29} - 8629787367894189712 p^{98} T^{30} + 34841884131333708 p^{105} T^{31} - 44226969761316 p^{112} T^{32} - 22301615936 p^{120} T^{33} + 14087432 p^{128} T^{34} - 5308 p^{136} T^{35} + p^{144} T^{36} \) | |
| 11 | \( 1 + 31556 T + 497890568 T^{2} + 7592398298612 T^{3} + 101896177789882233 T^{4} + \)\(86\!\cdots\!36\)\( T^{5} + \)\(52\!\cdots\!44\)\( T^{6} - \)\(10\!\cdots\!88\)\( T^{7} - \)\(49\!\cdots\!96\)\( T^{8} - \)\(60\!\cdots\!24\)\( T^{9} - \)\(39\!\cdots\!76\)\( T^{10} - \)\(14\!\cdots\!28\)\( T^{11} + \)\(60\!\cdots\!00\)\( T^{12} + \)\(15\!\cdots\!16\)\( T^{13} + \)\(26\!\cdots\!00\)\( T^{14} + \)\(51\!\cdots\!72\)\( T^{15} + \)\(49\!\cdots\!02\)\( T^{16} - \)\(66\!\cdots\!92\)\( p T^{17} - \)\(22\!\cdots\!72\)\( p^{2} T^{18} - \)\(66\!\cdots\!92\)\( p^{9} T^{19} + \)\(49\!\cdots\!02\)\( p^{16} T^{20} + \)\(51\!\cdots\!72\)\( p^{24} T^{21} + \)\(26\!\cdots\!00\)\( p^{32} T^{22} + \)\(15\!\cdots\!16\)\( p^{40} T^{23} + \)\(60\!\cdots\!00\)\( p^{48} T^{24} - \)\(14\!\cdots\!28\)\( p^{56} T^{25} - \)\(39\!\cdots\!76\)\( p^{64} T^{26} - \)\(60\!\cdots\!24\)\( p^{72} T^{27} - \)\(49\!\cdots\!96\)\( p^{80} T^{28} - \)\(10\!\cdots\!88\)\( p^{88} T^{29} + \)\(52\!\cdots\!44\)\( p^{96} T^{30} + \)\(86\!\cdots\!36\)\( p^{104} T^{31} + 101896177789882233 p^{112} T^{32} + 7592398298612 p^{120} T^{33} + 497890568 p^{128} T^{34} + 31556 p^{136} T^{35} + p^{144} T^{36} \) | |
| 17 | \( 1 - 79443299400 T^{2} + \)\(30\!\cdots\!44\)\( T^{4} - \)\(76\!\cdots\!50\)\( T^{6} + \)\(14\!\cdots\!44\)\( T^{8} - \)\(20\!\cdots\!24\)\( T^{10} + \)\(24\!\cdots\!91\)\( T^{12} - \)\(24\!\cdots\!84\)\( T^{14} + \)\(21\!\cdots\!64\)\( T^{16} - \)\(15\!\cdots\!56\)\( T^{18} + \)\(21\!\cdots\!64\)\( p^{16} T^{20} - \)\(24\!\cdots\!84\)\( p^{32} T^{22} + \)\(24\!\cdots\!91\)\( p^{48} T^{24} - \)\(20\!\cdots\!24\)\( p^{64} T^{26} + \)\(14\!\cdots\!44\)\( p^{80} T^{28} - \)\(76\!\cdots\!50\)\( p^{96} T^{30} + \)\(30\!\cdots\!44\)\( p^{112} T^{32} - 79443299400 p^{128} T^{34} + p^{144} T^{36} \) | |
| 19 | \( 1 - 100288 T + 5028841472 T^{2} - 4728262683462440 T^{3} + \)\(72\!\cdots\!93\)\( T^{4} + \)\(50\!\cdots\!56\)\( T^{5} + \)\(24\!\cdots\!76\)\( T^{6} - \)\(87\!\cdots\!20\)\( T^{7} - \)\(35\!\cdots\!92\)\( T^{8} + \)\(55\!\cdots\!56\)\( T^{9} - \)\(15\!\cdots\!16\)\( p T^{10} + \)\(11\!\cdots\!52\)\( T^{11} - \)\(15\!\cdots\!52\)\( T^{12} - \)\(58\!\cdots\!96\)\( T^{13} + \)\(12\!\cdots\!48\)\( p T^{14} + \)\(17\!\cdots\!16\)\( T^{15} + \)\(59\!\cdots\!26\)\( T^{16} - \)\(64\!\cdots\!64\)\( T^{17} + \)\(68\!\cdots\!40\)\( T^{18} - \)\(64\!\cdots\!64\)\( p^{8} T^{19} + \)\(59\!\cdots\!26\)\( p^{16} T^{20} + \)\(17\!\cdots\!16\)\( p^{24} T^{21} + \)\(12\!\cdots\!48\)\( p^{33} T^{22} - \)\(58\!\cdots\!96\)\( p^{40} T^{23} - \)\(15\!\cdots\!52\)\( p^{48} T^{24} + \)\(11\!\cdots\!52\)\( p^{56} T^{25} - \)\(15\!\cdots\!16\)\( p^{65} T^{26} + \)\(55\!\cdots\!56\)\( p^{72} T^{27} - \)\(35\!\cdots\!92\)\( p^{80} T^{28} - \)\(87\!\cdots\!20\)\( p^{88} T^{29} + \)\(24\!\cdots\!76\)\( p^{96} T^{30} + \)\(50\!\cdots\!56\)\( p^{104} T^{31} + \)\(72\!\cdots\!93\)\( p^{112} T^{32} - 4728262683462440 p^{120} T^{33} + 5028841472 p^{128} T^{34} - 100288 p^{136} T^{35} + p^{144} T^{36} \) | |
| 23 | \( 1 - 969313822206 T^{2} + \)\(44\!\cdots\!37\)\( T^{4} - \)\(12\!\cdots\!56\)\( T^{6} + \)\(25\!\cdots\!08\)\( T^{8} - \)\(36\!\cdots\!48\)\( T^{10} + \)\(40\!\cdots\!92\)\( T^{12} - \)\(36\!\cdots\!88\)\( T^{14} + \)\(28\!\cdots\!98\)\( T^{16} - \)\(21\!\cdots\!92\)\( T^{18} + \)\(28\!\cdots\!98\)\( p^{16} T^{20} - \)\(36\!\cdots\!88\)\( p^{32} T^{22} + \)\(40\!\cdots\!92\)\( p^{48} T^{24} - \)\(36\!\cdots\!48\)\( p^{64} T^{26} + \)\(25\!\cdots\!08\)\( p^{80} T^{28} - \)\(12\!\cdots\!56\)\( p^{96} T^{30} + \)\(44\!\cdots\!37\)\( p^{112} T^{32} - 969313822206 p^{128} T^{34} + p^{144} T^{36} \) | |
| 29 | \( ( 1 + 1239512 T + 2527783467579 T^{2} + 1577419580838429880 T^{3} + \)\(20\!\cdots\!06\)\( T^{4} + \)\(60\!\cdots\!04\)\( T^{5} + \)\(11\!\cdots\!34\)\( T^{6} + \)\(39\!\cdots\!08\)\( T^{7} + \)\(89\!\cdots\!16\)\( T^{8} + \)\(33\!\cdots\!84\)\( T^{9} + \)\(89\!\cdots\!16\)\( p^{8} T^{10} + \)\(39\!\cdots\!08\)\( p^{16} T^{11} + \)\(11\!\cdots\!34\)\( p^{24} T^{12} + \)\(60\!\cdots\!04\)\( p^{32} T^{13} + \)\(20\!\cdots\!06\)\( p^{40} T^{14} + 1577419580838429880 p^{48} T^{15} + 2527783467579 p^{56} T^{16} + 1239512 p^{64} T^{17} + p^{72} T^{18} )^{2} \) | |
| 31 | \( 1 + 1892664 T + 1791088508448 T^{2} + 609436150027888944 T^{3} + \)\(14\!\cdots\!41\)\( T^{4} + \)\(31\!\cdots\!28\)\( T^{5} + \)\(34\!\cdots\!92\)\( T^{6} + \)\(13\!\cdots\!56\)\( T^{7} + \)\(19\!\cdots\!40\)\( T^{8} + \)\(39\!\cdots\!84\)\( T^{9} + \)\(47\!\cdots\!40\)\( T^{10} + \)\(16\!\cdots\!56\)\( T^{11} + \)\(12\!\cdots\!76\)\( T^{12} + \)\(34\!\cdots\!76\)\( T^{13} + \)\(49\!\cdots\!80\)\( T^{14} + \)\(21\!\cdots\!76\)\( T^{15} + \)\(10\!\cdots\!46\)\( T^{16} + \)\(23\!\cdots\!68\)\( T^{17} + \)\(39\!\cdots\!08\)\( T^{18} + \)\(23\!\cdots\!68\)\( p^{8} T^{19} + \)\(10\!\cdots\!46\)\( p^{16} T^{20} + \)\(21\!\cdots\!76\)\( p^{24} T^{21} + \)\(49\!\cdots\!80\)\( p^{32} T^{22} + \)\(34\!\cdots\!76\)\( p^{40} T^{23} + \)\(12\!\cdots\!76\)\( p^{48} T^{24} + \)\(16\!\cdots\!56\)\( p^{56} T^{25} + \)\(47\!\cdots\!40\)\( p^{64} T^{26} + \)\(39\!\cdots\!84\)\( p^{72} T^{27} + \)\(19\!\cdots\!40\)\( p^{80} T^{28} + \)\(13\!\cdots\!56\)\( p^{88} T^{29} + \)\(34\!\cdots\!92\)\( p^{96} T^{30} + \)\(31\!\cdots\!28\)\( p^{104} T^{31} + \)\(14\!\cdots\!41\)\( p^{112} T^{32} + 609436150027888944 p^{120} T^{33} + 1791088508448 p^{128} T^{34} + 1892664 p^{136} T^{35} + p^{144} T^{36} \) | |
| 37 | \( 1 + 8343978 T + 34810984432242 T^{2} + \)\(10\!\cdots\!42\)\( T^{3} + \)\(28\!\cdots\!60\)\( T^{4} + \)\(64\!\cdots\!66\)\( T^{5} + \)\(12\!\cdots\!10\)\( T^{6} + \)\(22\!\cdots\!34\)\( T^{7} + \)\(33\!\cdots\!88\)\( T^{8} + \)\(32\!\cdots\!14\)\( T^{9} - \)\(10\!\cdots\!98\)\( T^{10} - \)\(15\!\cdots\!74\)\( T^{11} - \)\(47\!\cdots\!45\)\( T^{12} - \)\(10\!\cdots\!48\)\( T^{13} - \)\(18\!\cdots\!60\)\( T^{14} - \)\(28\!\cdots\!12\)\( T^{15} - \)\(38\!\cdots\!04\)\( T^{16} - \)\(54\!\cdots\!00\)\( T^{17} - \)\(93\!\cdots\!48\)\( T^{18} - \)\(54\!\cdots\!00\)\( p^{8} T^{19} - \)\(38\!\cdots\!04\)\( p^{16} T^{20} - \)\(28\!\cdots\!12\)\( p^{24} T^{21} - \)\(18\!\cdots\!60\)\( p^{32} T^{22} - \)\(10\!\cdots\!48\)\( p^{40} T^{23} - \)\(47\!\cdots\!45\)\( p^{48} T^{24} - \)\(15\!\cdots\!74\)\( p^{56} T^{25} - \)\(10\!\cdots\!98\)\( p^{64} T^{26} + \)\(32\!\cdots\!14\)\( p^{72} T^{27} + \)\(33\!\cdots\!88\)\( p^{80} T^{28} + \)\(22\!\cdots\!34\)\( p^{88} T^{29} + \)\(12\!\cdots\!10\)\( p^{96} T^{30} + \)\(64\!\cdots\!66\)\( p^{104} T^{31} + \)\(28\!\cdots\!60\)\( p^{112} T^{32} + \)\(10\!\cdots\!42\)\( p^{120} T^{33} + 34810984432242 p^{128} T^{34} + 8343978 p^{136} T^{35} + p^{144} T^{36} \) | |
| 41 | \( 1 - 1140178 T + 650002935842 T^{2} - 21941460521330260498 T^{3} + \)\(58\!\cdots\!21\)\( T^{4} + \)\(20\!\cdots\!28\)\( T^{5} - \)\(34\!\cdots\!64\)\( T^{6} + \)\(11\!\cdots\!56\)\( T^{7} - \)\(60\!\cdots\!44\)\( T^{8} + \)\(36\!\cdots\!92\)\( T^{9} - \)\(91\!\cdots\!80\)\( T^{10} + \)\(72\!\cdots\!00\)\( T^{11} + \)\(44\!\cdots\!88\)\( T^{12} - \)\(79\!\cdots\!12\)\( T^{13} + \)\(25\!\cdots\!48\)\( T^{14} - \)\(57\!\cdots\!80\)\( T^{15} - \)\(41\!\cdots\!50\)\( T^{16} + \)\(50\!\cdots\!40\)\( T^{17} - \)\(12\!\cdots\!80\)\( T^{18} + \)\(50\!\cdots\!40\)\( p^{8} T^{19} - \)\(41\!\cdots\!50\)\( p^{16} T^{20} - \)\(57\!\cdots\!80\)\( p^{24} T^{21} + \)\(25\!\cdots\!48\)\( p^{32} T^{22} - \)\(79\!\cdots\!12\)\( p^{40} T^{23} + \)\(44\!\cdots\!88\)\( p^{48} T^{24} + \)\(72\!\cdots\!00\)\( p^{56} T^{25} - \)\(91\!\cdots\!80\)\( p^{64} T^{26} + \)\(36\!\cdots\!92\)\( p^{72} T^{27} - \)\(60\!\cdots\!44\)\( p^{80} T^{28} + \)\(11\!\cdots\!56\)\( p^{88} T^{29} - \)\(34\!\cdots\!64\)\( p^{96} T^{30} + \)\(20\!\cdots\!28\)\( p^{104} T^{31} + \)\(58\!\cdots\!21\)\( p^{112} T^{32} - 21941460521330260498 p^{120} T^{33} + 650002935842 p^{128} T^{34} - 1140178 p^{136} T^{35} + p^{144} T^{36} \) | |
| 43 | \( 1 - 78194075403600 T^{2} + \)\(34\!\cdots\!24\)\( T^{4} - \)\(11\!\cdots\!06\)\( T^{6} + \)\(28\!\cdots\!08\)\( T^{8} - \)\(59\!\cdots\!44\)\( T^{10} + \)\(10\!\cdots\!79\)\( T^{12} - \)\(17\!\cdots\!72\)\( T^{14} + \)\(24\!\cdots\!24\)\( T^{16} - \)\(29\!\cdots\!84\)\( T^{18} + \)\(24\!\cdots\!24\)\( p^{16} T^{20} - \)\(17\!\cdots\!72\)\( p^{32} T^{22} + \)\(10\!\cdots\!79\)\( p^{48} T^{24} - \)\(59\!\cdots\!44\)\( p^{64} T^{26} + \)\(28\!\cdots\!08\)\( p^{80} T^{28} - \)\(11\!\cdots\!06\)\( p^{96} T^{30} + \)\(34\!\cdots\!24\)\( p^{112} T^{32} - 78194075403600 p^{128} T^{34} + p^{144} T^{36} \) | |
| 47 | \( 1 + 13368572 T + 89359358659592 T^{2} + \)\(40\!\cdots\!24\)\( T^{3} + \)\(20\!\cdots\!24\)\( T^{4} + \)\(14\!\cdots\!20\)\( T^{5} + \)\(95\!\cdots\!20\)\( T^{6} + \)\(46\!\cdots\!12\)\( T^{7} + \)\(19\!\cdots\!04\)\( T^{8} + \)\(78\!\cdots\!08\)\( T^{9} + \)\(32\!\cdots\!16\)\( T^{10} + \)\(11\!\cdots\!52\)\( T^{11} + \)\(53\!\cdots\!83\)\( T^{12} - \)\(30\!\cdots\!32\)\( T^{13} - \)\(26\!\cdots\!24\)\( T^{14} - \)\(15\!\cdots\!44\)\( T^{15} - \)\(88\!\cdots\!52\)\( T^{16} - \)\(52\!\cdots\!52\)\( T^{17} - \)\(28\!\cdots\!48\)\( T^{18} - \)\(52\!\cdots\!52\)\( p^{8} T^{19} - \)\(88\!\cdots\!52\)\( p^{16} T^{20} - \)\(15\!\cdots\!44\)\( p^{24} T^{21} - \)\(26\!\cdots\!24\)\( p^{32} T^{22} - \)\(30\!\cdots\!32\)\( p^{40} T^{23} + \)\(53\!\cdots\!83\)\( p^{48} T^{24} + \)\(11\!\cdots\!52\)\( p^{56} T^{25} + \)\(32\!\cdots\!16\)\( p^{64} T^{26} + \)\(78\!\cdots\!08\)\( p^{72} T^{27} + \)\(19\!\cdots\!04\)\( p^{80} T^{28} + \)\(46\!\cdots\!12\)\( p^{88} T^{29} + \)\(95\!\cdots\!20\)\( p^{96} T^{30} + \)\(14\!\cdots\!20\)\( p^{104} T^{31} + \)\(20\!\cdots\!24\)\( p^{112} T^{32} + \)\(40\!\cdots\!24\)\( p^{120} T^{33} + 89359358659592 p^{128} T^{34} + 13368572 p^{136} T^{35} + p^{144} T^{36} \) | |
| 53 | \( ( 1 - 25280674 T + 532214168618055 T^{2} - \)\(77\!\cdots\!68\)\( T^{3} + \)\(10\!\cdots\!18\)\( T^{4} - \)\(11\!\cdots\!08\)\( T^{5} + \)\(11\!\cdots\!50\)\( T^{6} - \)\(11\!\cdots\!04\)\( T^{7} + \)\(99\!\cdots\!76\)\( T^{8} - \)\(80\!\cdots\!00\)\( T^{9} + \)\(99\!\cdots\!76\)\( p^{8} T^{10} - \)\(11\!\cdots\!04\)\( p^{16} T^{11} + \)\(11\!\cdots\!50\)\( p^{24} T^{12} - \)\(11\!\cdots\!08\)\( p^{32} T^{13} + \)\(10\!\cdots\!18\)\( p^{40} T^{14} - \)\(77\!\cdots\!68\)\( p^{48} T^{15} + 532214168618055 p^{56} T^{16} - 25280674 p^{64} T^{17} + p^{72} T^{18} )^{2} \) | |
| 59 | \( 1 - 2127976 T + 2264140928288 T^{2} - \)\(28\!\cdots\!76\)\( T^{3} + \)\(58\!\cdots\!45\)\( T^{4} + \)\(37\!\cdots\!92\)\( T^{5} + \)\(30\!\cdots\!36\)\( T^{6} - \)\(13\!\cdots\!76\)\( T^{7} + \)\(61\!\cdots\!36\)\( T^{8} + \)\(21\!\cdots\!24\)\( T^{9} + \)\(21\!\cdots\!96\)\( T^{10} - \)\(21\!\cdots\!84\)\( T^{11} - \)\(15\!\cdots\!08\)\( T^{12} + \)\(50\!\cdots\!24\)\( T^{13} + \)\(74\!\cdots\!44\)\( T^{14} - \)\(37\!\cdots\!16\)\( T^{15} - \)\(68\!\cdots\!94\)\( T^{16} + \)\(64\!\cdots\!28\)\( T^{17} + \)\(18\!\cdots\!92\)\( T^{18} + \)\(64\!\cdots\!28\)\( p^{8} T^{19} - \)\(68\!\cdots\!94\)\( p^{16} T^{20} - \)\(37\!\cdots\!16\)\( p^{24} T^{21} + \)\(74\!\cdots\!44\)\( p^{32} T^{22} + \)\(50\!\cdots\!24\)\( p^{40} T^{23} - \)\(15\!\cdots\!08\)\( p^{48} T^{24} - \)\(21\!\cdots\!84\)\( p^{56} T^{25} + \)\(21\!\cdots\!96\)\( p^{64} T^{26} + \)\(21\!\cdots\!24\)\( p^{72} T^{27} + \)\(61\!\cdots\!36\)\( p^{80} T^{28} - \)\(13\!\cdots\!76\)\( p^{88} T^{29} + \)\(30\!\cdots\!36\)\( p^{96} T^{30} + \)\(37\!\cdots\!92\)\( p^{104} T^{31} + \)\(58\!\cdots\!45\)\( p^{112} T^{32} - \)\(28\!\cdots\!76\)\( p^{120} T^{33} + 2264140928288 p^{128} T^{34} - 2127976 p^{136} T^{35} + p^{144} T^{36} \) | |
| 61 | \( ( 1 + 26008258 T + 1362522568542033 T^{2} + \)\(30\!\cdots\!36\)\( T^{3} + \)\(88\!\cdots\!08\)\( T^{4} + \)\(16\!\cdots\!76\)\( T^{5} + \)\(35\!\cdots\!76\)\( T^{6} + \)\(56\!\cdots\!00\)\( T^{7} + \)\(96\!\cdots\!94\)\( T^{8} + \)\(12\!\cdots\!56\)\( T^{9} + \)\(96\!\cdots\!94\)\( p^{8} T^{10} + \)\(56\!\cdots\!00\)\( p^{16} T^{11} + \)\(35\!\cdots\!76\)\( p^{24} T^{12} + \)\(16\!\cdots\!76\)\( p^{32} T^{13} + \)\(88\!\cdots\!08\)\( p^{40} T^{14} + \)\(30\!\cdots\!36\)\( p^{48} T^{15} + 1362522568542033 p^{56} T^{16} + 26008258 p^{64} T^{17} + p^{72} T^{18} )^{2} \) | |
| 67 | \( 1 - 960292 T + 461080362632 T^{2} + \)\(44\!\cdots\!12\)\( T^{3} - \)\(25\!\cdots\!51\)\( T^{4} + \)\(14\!\cdots\!32\)\( T^{5} + \)\(10\!\cdots\!60\)\( T^{6} + \)\(15\!\cdots\!36\)\( T^{7} + \)\(67\!\cdots\!84\)\( T^{8} - \)\(18\!\cdots\!60\)\( T^{9} + \)\(33\!\cdots\!36\)\( T^{10} - \)\(13\!\cdots\!16\)\( p T^{11} + \)\(52\!\cdots\!88\)\( T^{12} - \)\(16\!\cdots\!76\)\( T^{13} + \)\(35\!\cdots\!36\)\( T^{14} - \)\(10\!\cdots\!60\)\( T^{15} - \)\(60\!\cdots\!82\)\( T^{16} + \)\(14\!\cdots\!00\)\( T^{17} - \)\(47\!\cdots\!88\)\( T^{18} + \)\(14\!\cdots\!00\)\( p^{8} T^{19} - \)\(60\!\cdots\!82\)\( p^{16} T^{20} - \)\(10\!\cdots\!60\)\( p^{24} T^{21} + \)\(35\!\cdots\!36\)\( p^{32} T^{22} - \)\(16\!\cdots\!76\)\( p^{40} T^{23} + \)\(52\!\cdots\!88\)\( p^{48} T^{24} - \)\(13\!\cdots\!16\)\( p^{57} T^{25} + \)\(33\!\cdots\!36\)\( p^{64} T^{26} - \)\(18\!\cdots\!60\)\( p^{72} T^{27} + \)\(67\!\cdots\!84\)\( p^{80} T^{28} + \)\(15\!\cdots\!36\)\( p^{88} T^{29} + \)\(10\!\cdots\!60\)\( p^{96} T^{30} + \)\(14\!\cdots\!32\)\( p^{104} T^{31} - \)\(25\!\cdots\!51\)\( p^{112} T^{32} + \)\(44\!\cdots\!12\)\( p^{120} T^{33} + 461080362632 p^{128} T^{34} - 960292 p^{136} T^{35} + p^{144} T^{36} \) | |
| 71 | \( 1 - 67412140 T + 2272198309689800 T^{2} - \)\(37\!\cdots\!72\)\( T^{3} + \)\(91\!\cdots\!76\)\( T^{4} - \)\(58\!\cdots\!36\)\( T^{5} + \)\(25\!\cdots\!32\)\( T^{6} - \)\(62\!\cdots\!48\)\( T^{7} + \)\(13\!\cdots\!04\)\( T^{8} - \)\(41\!\cdots\!40\)\( T^{9} + \)\(14\!\cdots\!72\)\( T^{10} - \)\(39\!\cdots\!08\)\( T^{11} + \)\(98\!\cdots\!47\)\( T^{12} - \)\(28\!\cdots\!96\)\( T^{13} + \)\(80\!\cdots\!08\)\( T^{14} - \)\(17\!\cdots\!44\)\( T^{15} + \)\(42\!\cdots\!08\)\( T^{16} - \)\(13\!\cdots\!28\)\( T^{17} + \)\(40\!\cdots\!28\)\( T^{18} - \)\(13\!\cdots\!28\)\( p^{8} T^{19} + \)\(42\!\cdots\!08\)\( p^{16} T^{20} - \)\(17\!\cdots\!44\)\( p^{24} T^{21} + \)\(80\!\cdots\!08\)\( p^{32} T^{22} - \)\(28\!\cdots\!96\)\( p^{40} T^{23} + \)\(98\!\cdots\!47\)\( p^{48} T^{24} - \)\(39\!\cdots\!08\)\( p^{56} T^{25} + \)\(14\!\cdots\!72\)\( p^{64} T^{26} - \)\(41\!\cdots\!40\)\( p^{72} T^{27} + \)\(13\!\cdots\!04\)\( p^{80} T^{28} - \)\(62\!\cdots\!48\)\( p^{88} T^{29} + \)\(25\!\cdots\!32\)\( p^{96} T^{30} - \)\(58\!\cdots\!36\)\( p^{104} T^{31} + \)\(91\!\cdots\!76\)\( p^{112} T^{32} - \)\(37\!\cdots\!72\)\( p^{120} T^{33} + 2272198309689800 p^{128} T^{34} - 67412140 p^{136} T^{35} + p^{144} T^{36} \) | |
| 73 | \( 1 + 145213226 T + 10543440502663538 T^{2} + \)\(54\!\cdots\!30\)\( T^{3} + \)\(25\!\cdots\!21\)\( T^{4} + \)\(11\!\cdots\!76\)\( T^{5} + \)\(51\!\cdots\!28\)\( T^{6} + \)\(28\!\cdots\!08\)\( p T^{7} + \)\(81\!\cdots\!16\)\( T^{8} + \)\(30\!\cdots\!12\)\( T^{9} + \)\(11\!\cdots\!24\)\( T^{10} + \)\(38\!\cdots\!84\)\( T^{11} + \)\(13\!\cdots\!56\)\( T^{12} + \)\(43\!\cdots\!32\)\( T^{13} + \)\(13\!\cdots\!12\)\( T^{14} + \)\(43\!\cdots\!68\)\( T^{15} + \)\(13\!\cdots\!06\)\( T^{16} + \)\(38\!\cdots\!08\)\( T^{17} + \)\(11\!\cdots\!76\)\( T^{18} + \)\(38\!\cdots\!08\)\( p^{8} T^{19} + \)\(13\!\cdots\!06\)\( p^{16} T^{20} + \)\(43\!\cdots\!68\)\( p^{24} T^{21} + \)\(13\!\cdots\!12\)\( p^{32} T^{22} + \)\(43\!\cdots\!32\)\( p^{40} T^{23} + \)\(13\!\cdots\!56\)\( p^{48} T^{24} + \)\(38\!\cdots\!84\)\( p^{56} T^{25} + \)\(11\!\cdots\!24\)\( p^{64} T^{26} + \)\(30\!\cdots\!12\)\( p^{72} T^{27} + \)\(81\!\cdots\!16\)\( p^{80} T^{28} + \)\(28\!\cdots\!08\)\( p^{89} T^{29} + \)\(51\!\cdots\!28\)\( p^{96} T^{30} + \)\(11\!\cdots\!76\)\( p^{104} T^{31} + \)\(25\!\cdots\!21\)\( p^{112} T^{32} + \)\(54\!\cdots\!30\)\( p^{120} T^{33} + 10543440502663538 p^{128} T^{34} + 145213226 p^{136} T^{35} + p^{144} T^{36} \) | |
| 79 | \( ( 1 + 38414560 T + 9112239695216607 T^{2} + \)\(31\!\cdots\!08\)\( T^{3} + \)\(40\!\cdots\!74\)\( T^{4} + \)\(12\!\cdots\!08\)\( T^{5} + \)\(11\!\cdots\!78\)\( T^{6} + \)\(33\!\cdots\!12\)\( T^{7} + \)\(24\!\cdots\!88\)\( T^{8} + \)\(59\!\cdots\!80\)\( T^{9} + \)\(24\!\cdots\!88\)\( p^{8} T^{10} + \)\(33\!\cdots\!12\)\( p^{16} T^{11} + \)\(11\!\cdots\!78\)\( p^{24} T^{12} + \)\(12\!\cdots\!08\)\( p^{32} T^{13} + \)\(40\!\cdots\!74\)\( p^{40} T^{14} + \)\(31\!\cdots\!08\)\( p^{48} T^{15} + 9112239695216607 p^{56} T^{16} + 38414560 p^{64} T^{17} + p^{72} T^{18} )^{2} \) | |
| 83 | \( 1 + 241951556 T + 29270277725410568 T^{2} + \)\(26\!\cdots\!48\)\( T^{3} + \)\(22\!\cdots\!09\)\( T^{4} + \)\(16\!\cdots\!64\)\( T^{5} + \)\(10\!\cdots\!24\)\( T^{6} + \)\(67\!\cdots\!32\)\( T^{7} + \)\(40\!\cdots\!40\)\( T^{8} + \)\(23\!\cdots\!88\)\( T^{9} + \)\(12\!\cdots\!76\)\( T^{10} + \)\(67\!\cdots\!96\)\( T^{11} + \)\(35\!\cdots\!76\)\( T^{12} + \)\(17\!\cdots\!92\)\( T^{13} + \)\(88\!\cdots\!92\)\( T^{14} + \)\(44\!\cdots\!24\)\( T^{15} + \)\(21\!\cdots\!14\)\( T^{16} + \)\(10\!\cdots\!60\)\( T^{17} + \)\(49\!\cdots\!00\)\( T^{18} + \)\(10\!\cdots\!60\)\( p^{8} T^{19} + \)\(21\!\cdots\!14\)\( p^{16} T^{20} + \)\(44\!\cdots\!24\)\( p^{24} T^{21} + \)\(88\!\cdots\!92\)\( p^{32} T^{22} + \)\(17\!\cdots\!92\)\( p^{40} T^{23} + \)\(35\!\cdots\!76\)\( p^{48} T^{24} + \)\(67\!\cdots\!96\)\( p^{56} T^{25} + \)\(12\!\cdots\!76\)\( p^{64} T^{26} + \)\(23\!\cdots\!88\)\( p^{72} T^{27} + \)\(40\!\cdots\!40\)\( p^{80} T^{28} + \)\(67\!\cdots\!32\)\( p^{88} T^{29} + \)\(10\!\cdots\!24\)\( p^{96} T^{30} + \)\(16\!\cdots\!64\)\( p^{104} T^{31} + \)\(22\!\cdots\!09\)\( p^{112} T^{32} + \)\(26\!\cdots\!48\)\( p^{120} T^{33} + 29270277725410568 p^{128} T^{34} + 241951556 p^{136} T^{35} + p^{144} T^{36} \) | |
| 89 | \( 1 + 89187110 T + 3977170295076050 T^{2} - \)\(44\!\cdots\!58\)\( T^{3} - \)\(33\!\cdots\!91\)\( T^{4} + \)\(47\!\cdots\!36\)\( T^{5} + \)\(27\!\cdots\!92\)\( T^{6} + \)\(10\!\cdots\!28\)\( T^{7} - \)\(74\!\cdots\!64\)\( T^{8} - \)\(91\!\cdots\!72\)\( T^{9} - \)\(46\!\cdots\!24\)\( T^{10} + \)\(30\!\cdots\!56\)\( T^{11} + \)\(16\!\cdots\!64\)\( T^{12} - \)\(64\!\cdots\!68\)\( T^{13} - \)\(63\!\cdots\!24\)\( T^{14} - \)\(64\!\cdots\!60\)\( T^{15} + \)\(19\!\cdots\!66\)\( T^{16} + \)\(39\!\cdots\!80\)\( T^{17} - \)\(28\!\cdots\!16\)\( T^{18} + \)\(39\!\cdots\!80\)\( p^{8} T^{19} + \)\(19\!\cdots\!66\)\( p^{16} T^{20} - \)\(64\!\cdots\!60\)\( p^{24} T^{21} - \)\(63\!\cdots\!24\)\( p^{32} T^{22} - \)\(64\!\cdots\!68\)\( p^{40} T^{23} + \)\(16\!\cdots\!64\)\( p^{48} T^{24} + \)\(30\!\cdots\!56\)\( p^{56} T^{25} - \)\(46\!\cdots\!24\)\( p^{64} T^{26} - \)\(91\!\cdots\!72\)\( p^{72} T^{27} - \)\(74\!\cdots\!64\)\( p^{80} T^{28} + \)\(10\!\cdots\!28\)\( p^{88} T^{29} + \)\(27\!\cdots\!92\)\( p^{96} T^{30} + \)\(47\!\cdots\!36\)\( p^{104} T^{31} - \)\(33\!\cdots\!91\)\( p^{112} T^{32} - \)\(44\!\cdots\!58\)\( p^{120} T^{33} + 3977170295076050 p^{128} T^{34} + 89187110 p^{136} T^{35} + p^{144} T^{36} \) | |
| 97 | \( 1 - 331183146 T + 54841138097228658 T^{2} - \)\(65\!\cdots\!54\)\( T^{3} + \)\(80\!\cdots\!09\)\( T^{4} - \)\(11\!\cdots\!88\)\( T^{5} + \)\(14\!\cdots\!84\)\( T^{6} - \)\(15\!\cdots\!60\)\( T^{7} + \)\(16\!\cdots\!60\)\( T^{8} - \)\(17\!\cdots\!80\)\( T^{9} + \)\(19\!\cdots\!56\)\( T^{10} - \)\(18\!\cdots\!72\)\( T^{11} + \)\(18\!\cdots\!96\)\( T^{12} - \)\(18\!\cdots\!56\)\( T^{13} + \)\(17\!\cdots\!92\)\( T^{14} - \)\(16\!\cdots\!24\)\( T^{15} + \)\(14\!\cdots\!94\)\( T^{16} - \)\(13\!\cdots\!60\)\( T^{17} + \)\(12\!\cdots\!00\)\( T^{18} - \)\(13\!\cdots\!60\)\( p^{8} T^{19} + \)\(14\!\cdots\!94\)\( p^{16} T^{20} - \)\(16\!\cdots\!24\)\( p^{24} T^{21} + \)\(17\!\cdots\!92\)\( p^{32} T^{22} - \)\(18\!\cdots\!56\)\( p^{40} T^{23} + \)\(18\!\cdots\!96\)\( p^{48} T^{24} - \)\(18\!\cdots\!72\)\( p^{56} T^{25} + \)\(19\!\cdots\!56\)\( p^{64} T^{26} - \)\(17\!\cdots\!80\)\( p^{72} T^{27} + \)\(16\!\cdots\!60\)\( p^{80} T^{28} - \)\(15\!\cdots\!60\)\( p^{88} T^{29} + \)\(14\!\cdots\!84\)\( p^{96} T^{30} - \)\(11\!\cdots\!88\)\( p^{104} T^{31} + \)\(80\!\cdots\!09\)\( p^{112} T^{32} - \)\(65\!\cdots\!54\)\( p^{120} T^{33} + 54841138097228658 p^{128} T^{34} - 331183146 p^{136} T^{35} + p^{144} T^{36} \) | |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{36} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−4.41492354785326048150456053023, −4.20135765660155903914263465666, −3.96583613947310360630415049569, −3.73679552969547901523796167446, −3.67281150933524152840610968049, −3.56982792856184175731776008990, −3.44127282609567687518556127624, −3.32000877804024169270182263908, −3.05408422724028792070986887905, −2.93170633778047652136089575625, −2.71932879136412965571663166493, −2.69640119852959524079412431704, −2.62430503859719021444994762903, −2.15940522878395711979312276157, −2.10158591761409269204667990133, −1.80055210888057831430773111256, −1.71755900986639415385785803681, −1.66800976048982456849415294158, −1.48455147725707155263913006686, −1.08813289762842012708914415989, −1.04085179613061025179548627336, −0.63221024970863912322397917411, −0.22917705208011738588258726606, −0.22360960336736790852513505208, −0.009419019266125920523290754123, 0.009419019266125920523290754123, 0.22360960336736790852513505208, 0.22917705208011738588258726606, 0.63221024970863912322397917411, 1.04085179613061025179548627336, 1.08813289762842012708914415989, 1.48455147725707155263913006686, 1.66800976048982456849415294158, 1.71755900986639415385785803681, 1.80055210888057831430773111256, 2.10158591761409269204667990133, 2.15940522878395711979312276157, 2.62430503859719021444994762903, 2.69640119852959524079412431704, 2.71932879136412965571663166493, 2.93170633778047652136089575625, 3.05408422724028792070986887905, 3.32000877804024169270182263908, 3.44127282609567687518556127624, 3.56982792856184175731776008990, 3.67281150933524152840610968049, 3.73679552969547901523796167446, 3.96583613947310360630415049569, 4.20135765660155903914263465666, 4.41492354785326048150456053023
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.