Dirichlet series
| L(s) = 1 | − 3·2-s − 93·3-s − 508·4-s + 438·5-s + 279·6-s + 922·7-s + 1.53e3·8-s + 1.33e4·9-s − 1.31e3·10-s − 2.86e4·11-s + 4.72e4·12-s + 1.68e3·13-s − 2.76e3·14-s − 4.07e4·15-s + 1.61e5·16-s − 3.99e4·18-s − 2.69e5·19-s − 2.22e5·20-s − 8.57e4·21-s + 8.60e4·22-s − 1.00e6·23-s − 1.42e5·24-s − 1.23e6·25-s − 5.05e3·26-s − 1.14e6·27-s − 4.68e5·28-s + 3.79e6·29-s + ⋯ |
| L(s) = 1 | − 0.187·2-s − 1.14·3-s − 1.98·4-s + 0.700·5-s + 0.215·6-s + 0.384·7-s + 0.374·8-s + 2.02·9-s − 0.131·10-s − 1.95·11-s + 2.27·12-s + 0.0589·13-s − 0.0720·14-s − 0.804·15-s + 2.46·16-s − 0.380·18-s − 2.06·19-s − 1.39·20-s − 0.440·21-s + 0.367·22-s − 3.57·23-s − 0.429·24-s − 3.17·25-s − 0.0110·26-s − 2.15·27-s − 0.762·28-s + 5.36·29-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{28}\right)^{s/2} \, \Gamma_{\C}(s)^{14} \, L(s)\cr=\mathstrut & \,\Lambda(9-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{28}\right)^{s/2} \, \Gamma_{\C}(s+4)^{14} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(28\) |
| Conductor: | \(3^{28}\) |
| Sign: | $1$ |
| Analytic conductor: | \(7.93177\times 10^{7}\) |
| Root analytic conductor: | \(1.91478\) |
| Motivic weight: | \(8\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((28,\ 3^{28} ,\ ( \ : [4]^{14} ),\ 1 )\) |
Particular Values
| \(L(\frac{9}{2})\) | \(\approx\) | \(0.4409342248\) |
| \(L(\frac12)\) | \(\approx\) | \(0.4409342248\) |
| \(L(5)\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 3 | \( 1 + 31 p T - 518 p^{2} T^{2} - 6503 p^{4} T^{3} + 34276 p^{6} T^{4} + 268033 p^{9} T^{5} - 368591 p^{12} T^{6} - 1590770 p^{16} T^{7} - 368591 p^{20} T^{8} + 268033 p^{25} T^{9} + 34276 p^{30} T^{10} - 6503 p^{36} T^{11} - 518 p^{42} T^{12} + 31 p^{49} T^{13} + p^{56} T^{14} \) |
| good | 2 | \( 1 + 3 T + 517 T^{2} + 771 p T^{3} + 50435 p T^{4} + 316305 p^{2} T^{5} + 1971755 p^{2} T^{6} + 7938237 p^{5} T^{7} - 6212203 p^{6} T^{8} - 157323315 p^{8} T^{9} + 2690747417 p^{8} T^{10} - 4452651 p^{14} T^{11} + 49992409581 p^{12} T^{12} + 90407817459 p^{16} T^{13} - 187771365237 p^{14} T^{14} + 90407817459 p^{24} T^{15} + 49992409581 p^{28} T^{16} - 4452651 p^{38} T^{17} + 2690747417 p^{40} T^{18} - 157323315 p^{48} T^{19} - 6212203 p^{54} T^{20} + 7938237 p^{61} T^{21} + 1971755 p^{66} T^{22} + 316305 p^{74} T^{23} + 50435 p^{81} T^{24} + 771 p^{89} T^{25} + 517 p^{96} T^{26} + 3 p^{104} T^{27} + p^{112} T^{28} \) |
| 5 | \( 1 - 438 T + 1430521 T^{2} - 598558974 T^{3} + 953033871901 T^{4} - 214918574105376 T^{5} + 332911391257961888 T^{6} + 15957556703449088808 p T^{7} + \)\(14\!\cdots\!08\)\( p^{2} T^{8} + \)\(83\!\cdots\!32\)\( p^{3} T^{9} - \)\(50\!\cdots\!76\)\( p^{4} T^{10} + \)\(14\!\cdots\!28\)\( p^{5} T^{11} + \)\(51\!\cdots\!98\)\( p^{6} T^{12} + \)\(15\!\cdots\!72\)\( p^{7} T^{13} + \)\(15\!\cdots\!78\)\( p^{8} T^{14} + \)\(15\!\cdots\!72\)\( p^{15} T^{15} + \)\(51\!\cdots\!98\)\( p^{22} T^{16} + \)\(14\!\cdots\!28\)\( p^{29} T^{17} - \)\(50\!\cdots\!76\)\( p^{36} T^{18} + \)\(83\!\cdots\!32\)\( p^{43} T^{19} + \)\(14\!\cdots\!08\)\( p^{50} T^{20} + 15957556703449088808 p^{57} T^{21} + 332911391257961888 p^{64} T^{22} - 214918574105376 p^{72} T^{23} + 953033871901 p^{80} T^{24} - 598558974 p^{88} T^{25} + 1430521 p^{96} T^{26} - 438 p^{104} T^{27} + p^{112} T^{28} \) | |
| 7 | \( 1 - 922 T - 17338443 T^{2} + 6448042614 p T^{3} + 2324648817741 p^{2} T^{4} - 1748430783475548 p^{3} T^{5} + 20677143394491348 p^{4} T^{6} + \)\(24\!\cdots\!48\)\( p^{5} T^{7} - \)\(48\!\cdots\!60\)\( p^{6} T^{8} - \)\(26\!\cdots\!96\)\( p^{8} T^{9} + \)\(79\!\cdots\!12\)\( p^{8} T^{10} + \)\(52\!\cdots\!92\)\( p^{9} T^{11} - \)\(75\!\cdots\!30\)\( p^{10} T^{12} + \)\(14\!\cdots\!04\)\( p^{12} T^{13} + \)\(68\!\cdots\!66\)\( p^{12} T^{14} + \)\(14\!\cdots\!04\)\( p^{20} T^{15} - \)\(75\!\cdots\!30\)\( p^{26} T^{16} + \)\(52\!\cdots\!92\)\( p^{33} T^{17} + \)\(79\!\cdots\!12\)\( p^{40} T^{18} - \)\(26\!\cdots\!96\)\( p^{48} T^{19} - \)\(48\!\cdots\!60\)\( p^{54} T^{20} + \)\(24\!\cdots\!48\)\( p^{61} T^{21} + 20677143394491348 p^{68} T^{22} - 1748430783475548 p^{75} T^{23} + 2324648817741 p^{82} T^{24} + 6448042614 p^{89} T^{25} - 17338443 p^{96} T^{26} - 922 p^{104} T^{27} + p^{112} T^{28} \) | |
| 11 | \( 1 + 237 p^{2} T + 114469697 p T^{2} + 21223204008 p^{3} T^{3} + 728211895538261056 T^{4} + \)\(13\!\cdots\!18\)\( T^{5} + \)\(23\!\cdots\!85\)\( p T^{6} + \)\(38\!\cdots\!83\)\( T^{7} + \)\(59\!\cdots\!24\)\( T^{8} + \)\(67\!\cdots\!01\)\( T^{9} + \)\(81\!\cdots\!25\)\( T^{10} + \)\(44\!\cdots\!62\)\( T^{11} + \)\(19\!\cdots\!53\)\( p T^{12} - \)\(10\!\cdots\!49\)\( T^{13} - \)\(11\!\cdots\!22\)\( T^{14} - \)\(10\!\cdots\!49\)\( p^{8} T^{15} + \)\(19\!\cdots\!53\)\( p^{17} T^{16} + \)\(44\!\cdots\!62\)\( p^{24} T^{17} + \)\(81\!\cdots\!25\)\( p^{32} T^{18} + \)\(67\!\cdots\!01\)\( p^{40} T^{19} + \)\(59\!\cdots\!24\)\( p^{48} T^{20} + \)\(38\!\cdots\!83\)\( p^{56} T^{21} + \)\(23\!\cdots\!85\)\( p^{65} T^{22} + \)\(13\!\cdots\!18\)\( p^{72} T^{23} + 728211895538261056 p^{80} T^{24} + 21223204008 p^{91} T^{25} + 114469697 p^{97} T^{26} + 237 p^{106} T^{27} + p^{112} T^{28} \) | |
| 13 | \( 1 - 1684 T - 3239027709 T^{2} + 56081547895548 T^{3} + 5079221844706551261 T^{4} - \)\(14\!\cdots\!28\)\( T^{5} - \)\(39\!\cdots\!00\)\( T^{6} + \)\(15\!\cdots\!84\)\( T^{7} + \)\(16\!\cdots\!00\)\( T^{8} - \)\(66\!\cdots\!68\)\( T^{9} - \)\(16\!\cdots\!60\)\( T^{10} - \)\(24\!\cdots\!36\)\( T^{11} + \)\(37\!\cdots\!38\)\( T^{12} + \)\(22\!\cdots\!52\)\( T^{13} - \)\(42\!\cdots\!02\)\( T^{14} + \)\(22\!\cdots\!52\)\( p^{8} T^{15} + \)\(37\!\cdots\!38\)\( p^{16} T^{16} - \)\(24\!\cdots\!36\)\( p^{24} T^{17} - \)\(16\!\cdots\!60\)\( p^{32} T^{18} - \)\(66\!\cdots\!68\)\( p^{40} T^{19} + \)\(16\!\cdots\!00\)\( p^{48} T^{20} + \)\(15\!\cdots\!84\)\( p^{56} T^{21} - \)\(39\!\cdots\!00\)\( p^{64} T^{22} - \)\(14\!\cdots\!28\)\( p^{72} T^{23} + 5079221844706551261 p^{80} T^{24} + 56081547895548 p^{88} T^{25} - 3239027709 p^{96} T^{26} - 1684 p^{104} T^{27} + p^{112} T^{28} \) | |
| 17 | \( 1 - 130044893 p^{2} T^{2} + \)\(74\!\cdots\!58\)\( T^{4} - \)\(10\!\cdots\!41\)\( T^{6} + \)\(11\!\cdots\!64\)\( T^{8} - \)\(98\!\cdots\!87\)\( T^{10} + \)\(76\!\cdots\!49\)\( T^{12} - \)\(54\!\cdots\!06\)\( T^{14} + \)\(76\!\cdots\!49\)\( p^{16} T^{16} - \)\(98\!\cdots\!87\)\( p^{32} T^{18} + \)\(11\!\cdots\!64\)\( p^{48} T^{20} - \)\(10\!\cdots\!41\)\( p^{64} T^{22} + \)\(74\!\cdots\!58\)\( p^{80} T^{24} - 130044893 p^{98} T^{26} + p^{112} T^{28} \) | |
| 19 | \( ( 1 + 134815 T + 49841728210 T^{2} + 2477583914167247 T^{3} + \)\(99\!\cdots\!88\)\( T^{4} + \)\(50\!\cdots\!77\)\( T^{5} + \)\(26\!\cdots\!89\)\( T^{6} + \)\(17\!\cdots\!18\)\( T^{7} + \)\(26\!\cdots\!89\)\( p^{8} T^{8} + \)\(50\!\cdots\!77\)\( p^{16} T^{9} + \)\(99\!\cdots\!88\)\( p^{24} T^{10} + 2477583914167247 p^{32} T^{11} + 49841728210 p^{40} T^{12} + 134815 p^{48} T^{13} + p^{56} T^{14} )^{2} \) | |
| 23 | \( 1 + 1000452 T + 962072243827 T^{2} + 628721562813094668 T^{3} + \)\(16\!\cdots\!15\)\( p T^{4} + \)\(20\!\cdots\!80\)\( T^{5} + \)\(97\!\cdots\!80\)\( T^{6} + \)\(42\!\cdots\!96\)\( T^{7} + \)\(17\!\cdots\!88\)\( T^{8} + \)\(65\!\cdots\!60\)\( T^{9} + \)\(23\!\cdots\!52\)\( T^{10} + \)\(78\!\cdots\!64\)\( T^{11} + \)\(25\!\cdots\!26\)\( T^{12} + \)\(75\!\cdots\!96\)\( T^{13} + \)\(21\!\cdots\!62\)\( T^{14} + \)\(75\!\cdots\!96\)\( p^{8} T^{15} + \)\(25\!\cdots\!26\)\( p^{16} T^{16} + \)\(78\!\cdots\!64\)\( p^{24} T^{17} + \)\(23\!\cdots\!52\)\( p^{32} T^{18} + \)\(65\!\cdots\!60\)\( p^{40} T^{19} + \)\(17\!\cdots\!88\)\( p^{48} T^{20} + \)\(42\!\cdots\!96\)\( p^{56} T^{21} + \)\(97\!\cdots\!80\)\( p^{64} T^{22} + \)\(20\!\cdots\!80\)\( p^{72} T^{23} + \)\(16\!\cdots\!15\)\( p^{81} T^{24} + 628721562813094668 p^{88} T^{25} + 962072243827 p^{96} T^{26} + 1000452 p^{104} T^{27} + p^{112} T^{28} \) | |
| 29 | \( 1 - 3797682 T + 9651920526409 T^{2} - 18397709688187751082 T^{3} + \)\(29\!\cdots\!25\)\( T^{4} - \)\(40\!\cdots\!40\)\( T^{5} + \)\(50\!\cdots\!56\)\( T^{6} - \)\(56\!\cdots\!76\)\( T^{7} + \)\(58\!\cdots\!84\)\( T^{8} - \)\(56\!\cdots\!56\)\( T^{9} + \)\(51\!\cdots\!52\)\( T^{10} - \)\(44\!\cdots\!60\)\( T^{11} + \)\(35\!\cdots\!42\)\( T^{12} - \)\(27\!\cdots\!36\)\( T^{13} + \)\(19\!\cdots\!78\)\( T^{14} - \)\(27\!\cdots\!36\)\( p^{8} T^{15} + \)\(35\!\cdots\!42\)\( p^{16} T^{16} - \)\(44\!\cdots\!60\)\( p^{24} T^{17} + \)\(51\!\cdots\!52\)\( p^{32} T^{18} - \)\(56\!\cdots\!56\)\( p^{40} T^{19} + \)\(58\!\cdots\!84\)\( p^{48} T^{20} - \)\(56\!\cdots\!76\)\( p^{56} T^{21} + \)\(50\!\cdots\!56\)\( p^{64} T^{22} - \)\(40\!\cdots\!40\)\( p^{72} T^{23} + \)\(29\!\cdots\!25\)\( p^{80} T^{24} - 18397709688187751082 p^{88} T^{25} + 9651920526409 p^{96} T^{26} - 3797682 p^{104} T^{27} + p^{112} T^{28} \) | |
| 31 | \( 1 + 164132 T - 3383827130283 T^{2} - 2431470882982368948 T^{3} + \)\(57\!\cdots\!17\)\( T^{4} + \)\(67\!\cdots\!00\)\( T^{5} - \)\(49\!\cdots\!68\)\( T^{6} - \)\(10\!\cdots\!04\)\( T^{7} + \)\(76\!\cdots\!48\)\( T^{8} + \)\(10\!\cdots\!76\)\( T^{9} + \)\(39\!\cdots\!96\)\( T^{10} - \)\(71\!\cdots\!44\)\( T^{11} - \)\(64\!\cdots\!14\)\( T^{12} + \)\(22\!\cdots\!52\)\( T^{13} + \)\(64\!\cdots\!58\)\( T^{14} + \)\(22\!\cdots\!52\)\( p^{8} T^{15} - \)\(64\!\cdots\!14\)\( p^{16} T^{16} - \)\(71\!\cdots\!44\)\( p^{24} T^{17} + \)\(39\!\cdots\!96\)\( p^{32} T^{18} + \)\(10\!\cdots\!76\)\( p^{40} T^{19} + \)\(76\!\cdots\!48\)\( p^{48} T^{20} - \)\(10\!\cdots\!04\)\( p^{56} T^{21} - \)\(49\!\cdots\!68\)\( p^{64} T^{22} + \)\(67\!\cdots\!00\)\( p^{72} T^{23} + \)\(57\!\cdots\!17\)\( p^{80} T^{24} - 2431470882982368948 p^{88} T^{25} - 3383827130283 p^{96} T^{26} + 164132 p^{104} T^{27} + p^{112} T^{28} \) | |
| 37 | \( ( 1 + 835834 T + 15202365322231 T^{2} + 15152681160847785884 T^{3} + \)\(11\!\cdots\!89\)\( T^{4} + \)\(11\!\cdots\!86\)\( T^{5} + \)\(60\!\cdots\!39\)\( T^{6} + \)\(53\!\cdots\!72\)\( T^{7} + \)\(60\!\cdots\!39\)\( p^{8} T^{8} + \)\(11\!\cdots\!86\)\( p^{16} T^{9} + \)\(11\!\cdots\!89\)\( p^{24} T^{10} + 15152681160847785884 p^{32} T^{11} + 15202365322231 p^{40} T^{12} + 835834 p^{48} T^{13} + p^{56} T^{14} )^{2} \) | |
| 41 | \( 1 + 10239447 T + 71419549807789 T^{2} + \)\(37\!\cdots\!42\)\( T^{3} + \)\(15\!\cdots\!88\)\( T^{4} + \)\(50\!\cdots\!52\)\( T^{5} + \)\(12\!\cdots\!57\)\( T^{6} + \)\(20\!\cdots\!93\)\( T^{7} - \)\(89\!\cdots\!88\)\( T^{8} - \)\(20\!\cdots\!41\)\( T^{9} - \)\(88\!\cdots\!13\)\( T^{10} - \)\(25\!\cdots\!24\)\( T^{11} - \)\(55\!\cdots\!65\)\( T^{12} - \)\(10\!\cdots\!57\)\( T^{13} - \)\(22\!\cdots\!82\)\( T^{14} - \)\(10\!\cdots\!57\)\( p^{8} T^{15} - \)\(55\!\cdots\!65\)\( p^{16} T^{16} - \)\(25\!\cdots\!24\)\( p^{24} T^{17} - \)\(88\!\cdots\!13\)\( p^{32} T^{18} - \)\(20\!\cdots\!41\)\( p^{40} T^{19} - \)\(89\!\cdots\!88\)\( p^{48} T^{20} + \)\(20\!\cdots\!93\)\( p^{56} T^{21} + \)\(12\!\cdots\!57\)\( p^{64} T^{22} + \)\(50\!\cdots\!52\)\( p^{72} T^{23} + \)\(15\!\cdots\!88\)\( p^{80} T^{24} + \)\(37\!\cdots\!42\)\( p^{88} T^{25} + 71419549807789 p^{96} T^{26} + 10239447 p^{104} T^{27} + p^{112} T^{28} \) | |
| 43 | \( 1 - 791815 T - 62186405173917 T^{2} + 20513384410334125692 T^{3} + \)\(20\!\cdots\!00\)\( T^{4} - \)\(66\!\cdots\!94\)\( T^{5} - \)\(49\!\cdots\!81\)\( T^{6} - \)\(31\!\cdots\!65\)\( T^{7} + \)\(94\!\cdots\!00\)\( T^{8} + \)\(55\!\cdots\!49\)\( T^{9} - \)\(15\!\cdots\!59\)\( T^{10} - \)\(38\!\cdots\!82\)\( T^{11} + \)\(22\!\cdots\!99\)\( T^{12} + \)\(79\!\cdots\!07\)\( T^{13} - \)\(27\!\cdots\!86\)\( T^{14} + \)\(79\!\cdots\!07\)\( p^{8} T^{15} + \)\(22\!\cdots\!99\)\( p^{16} T^{16} - \)\(38\!\cdots\!82\)\( p^{24} T^{17} - \)\(15\!\cdots\!59\)\( p^{32} T^{18} + \)\(55\!\cdots\!49\)\( p^{40} T^{19} + \)\(94\!\cdots\!00\)\( p^{48} T^{20} - \)\(31\!\cdots\!65\)\( p^{56} T^{21} - \)\(49\!\cdots\!81\)\( p^{64} T^{22} - \)\(66\!\cdots\!94\)\( p^{72} T^{23} + \)\(20\!\cdots\!00\)\( p^{80} T^{24} + 20513384410334125692 p^{88} T^{25} - 62186405173917 p^{96} T^{26} - 791815 p^{104} T^{27} + p^{112} T^{28} \) | |
| 47 | \( 1 - 31148628 T + 586258762922491 T^{2} - \)\(81\!\cdots\!64\)\( T^{3} + \)\(94\!\cdots\!29\)\( T^{4} - \)\(94\!\cdots\!16\)\( T^{5} + \)\(83\!\cdots\!00\)\( T^{6} - \)\(67\!\cdots\!00\)\( T^{7} + \)\(49\!\cdots\!40\)\( T^{8} - \)\(34\!\cdots\!60\)\( T^{9} + \)\(21\!\cdots\!80\)\( T^{10} - \)\(13\!\cdots\!24\)\( T^{11} + \)\(15\!\cdots\!30\)\( p T^{12} - \)\(39\!\cdots\!48\)\( T^{13} + \)\(19\!\cdots\!38\)\( T^{14} - \)\(39\!\cdots\!48\)\( p^{8} T^{15} + \)\(15\!\cdots\!30\)\( p^{17} T^{16} - \)\(13\!\cdots\!24\)\( p^{24} T^{17} + \)\(21\!\cdots\!80\)\( p^{32} T^{18} - \)\(34\!\cdots\!60\)\( p^{40} T^{19} + \)\(49\!\cdots\!40\)\( p^{48} T^{20} - \)\(67\!\cdots\!00\)\( p^{56} T^{21} + \)\(83\!\cdots\!00\)\( p^{64} T^{22} - \)\(94\!\cdots\!16\)\( p^{72} T^{23} + \)\(94\!\cdots\!29\)\( p^{80} T^{24} - \)\(81\!\cdots\!64\)\( p^{88} T^{25} + 586258762922491 p^{96} T^{26} - 31148628 p^{104} T^{27} + p^{112} T^{28} \) | |
| 53 | \( 1 - 322932119080262 T^{2} + \)\(59\!\cdots\!19\)\( T^{4} - \)\(76\!\cdots\!44\)\( T^{6} + \)\(77\!\cdots\!77\)\( T^{8} - \)\(64\!\cdots\!66\)\( T^{10} + \)\(46\!\cdots\!43\)\( T^{12} - \)\(30\!\cdots\!56\)\( T^{14} + \)\(46\!\cdots\!43\)\( p^{16} T^{16} - \)\(64\!\cdots\!66\)\( p^{32} T^{18} + \)\(77\!\cdots\!77\)\( p^{48} T^{20} - \)\(76\!\cdots\!44\)\( p^{64} T^{22} + \)\(59\!\cdots\!19\)\( p^{80} T^{24} - 322932119080262 p^{96} T^{26} + p^{112} T^{28} \) | |
| 59 | \( 1 + 83493795 T + 4028836844427487 T^{2} + \)\(14\!\cdots\!40\)\( T^{3} + \)\(40\!\cdots\!80\)\( T^{4} + \)\(98\!\cdots\!06\)\( T^{5} + \)\(20\!\cdots\!39\)\( T^{6} + \)\(39\!\cdots\!89\)\( T^{7} + \)\(69\!\cdots\!48\)\( T^{8} + \)\(11\!\cdots\!59\)\( T^{9} + \)\(17\!\cdots\!61\)\( T^{10} + \)\(24\!\cdots\!34\)\( T^{11} + \)\(34\!\cdots\!39\)\( T^{12} + \)\(44\!\cdots\!17\)\( T^{13} + \)\(55\!\cdots\!90\)\( T^{14} + \)\(44\!\cdots\!17\)\( p^{8} T^{15} + \)\(34\!\cdots\!39\)\( p^{16} T^{16} + \)\(24\!\cdots\!34\)\( p^{24} T^{17} + \)\(17\!\cdots\!61\)\( p^{32} T^{18} + \)\(11\!\cdots\!59\)\( p^{40} T^{19} + \)\(69\!\cdots\!48\)\( p^{48} T^{20} + \)\(39\!\cdots\!89\)\( p^{56} T^{21} + \)\(20\!\cdots\!39\)\( p^{64} T^{22} + \)\(98\!\cdots\!06\)\( p^{72} T^{23} + \)\(40\!\cdots\!80\)\( p^{80} T^{24} + \)\(14\!\cdots\!40\)\( p^{88} T^{25} + 4028836844427487 p^{96} T^{26} + 83493795 p^{104} T^{27} + p^{112} T^{28} \) | |
| 61 | \( 1 + 5255600 T - 915230344307973 T^{2} + \)\(17\!\cdots\!44\)\( T^{3} + \)\(46\!\cdots\!53\)\( T^{4} - \)\(27\!\cdots\!60\)\( T^{5} - \)\(15\!\cdots\!28\)\( T^{6} + \)\(11\!\cdots\!96\)\( T^{7} + \)\(37\!\cdots\!16\)\( T^{8} - \)\(26\!\cdots\!92\)\( T^{9} - \)\(85\!\cdots\!52\)\( T^{10} + \)\(32\!\cdots\!60\)\( T^{11} + \)\(19\!\cdots\!66\)\( T^{12} - \)\(17\!\cdots\!68\)\( T^{13} - \)\(40\!\cdots\!86\)\( T^{14} - \)\(17\!\cdots\!68\)\( p^{8} T^{15} + \)\(19\!\cdots\!66\)\( p^{16} T^{16} + \)\(32\!\cdots\!60\)\( p^{24} T^{17} - \)\(85\!\cdots\!52\)\( p^{32} T^{18} - \)\(26\!\cdots\!92\)\( p^{40} T^{19} + \)\(37\!\cdots\!16\)\( p^{48} T^{20} + \)\(11\!\cdots\!96\)\( p^{56} T^{21} - \)\(15\!\cdots\!28\)\( p^{64} T^{22} - \)\(27\!\cdots\!60\)\( p^{72} T^{23} + \)\(46\!\cdots\!53\)\( p^{80} T^{24} + \)\(17\!\cdots\!44\)\( p^{88} T^{25} - 915230344307973 p^{96} T^{26} + 5255600 p^{104} T^{27} + p^{112} T^{28} \) | |
| 67 | \( 1 + 8288855 T - 1999239595504017 T^{2} - \)\(80\!\cdots\!12\)\( T^{3} + \)\(21\!\cdots\!00\)\( T^{4} + \)\(19\!\cdots\!74\)\( T^{5} - \)\(16\!\cdots\!41\)\( T^{6} + \)\(57\!\cdots\!45\)\( T^{7} + \)\(10\!\cdots\!20\)\( T^{8} - \)\(63\!\cdots\!69\)\( T^{9} - \)\(55\!\cdots\!79\)\( T^{10} + \)\(33\!\cdots\!42\)\( T^{11} + \)\(26\!\cdots\!79\)\( T^{12} - \)\(71\!\cdots\!07\)\( T^{13} - \)\(11\!\cdots\!26\)\( T^{14} - \)\(71\!\cdots\!07\)\( p^{8} T^{15} + \)\(26\!\cdots\!79\)\( p^{16} T^{16} + \)\(33\!\cdots\!42\)\( p^{24} T^{17} - \)\(55\!\cdots\!79\)\( p^{32} T^{18} - \)\(63\!\cdots\!69\)\( p^{40} T^{19} + \)\(10\!\cdots\!20\)\( p^{48} T^{20} + \)\(57\!\cdots\!45\)\( p^{56} T^{21} - \)\(16\!\cdots\!41\)\( p^{64} T^{22} + \)\(19\!\cdots\!74\)\( p^{72} T^{23} + \)\(21\!\cdots\!00\)\( p^{80} T^{24} - \)\(80\!\cdots\!12\)\( p^{88} T^{25} - 1999239595504017 p^{96} T^{26} + 8288855 p^{104} T^{27} + p^{112} T^{28} \) | |
| 71 | \( 1 - 5530227942584786 T^{2} + \)\(15\!\cdots\!59\)\( T^{4} - \)\(28\!\cdots\!92\)\( T^{6} + \)\(39\!\cdots\!33\)\( T^{8} - \)\(43\!\cdots\!26\)\( T^{10} + \)\(37\!\cdots\!71\)\( T^{12} - \)\(26\!\cdots\!40\)\( T^{14} + \)\(37\!\cdots\!71\)\( p^{16} T^{16} - \)\(43\!\cdots\!26\)\( p^{32} T^{18} + \)\(39\!\cdots\!33\)\( p^{48} T^{20} - \)\(28\!\cdots\!92\)\( p^{64} T^{22} + \)\(15\!\cdots\!59\)\( p^{80} T^{24} - 5530227942584786 p^{96} T^{26} + p^{112} T^{28} \) | |
| 73 | \( ( 1 + 18360841 T + 3410148934050418 T^{2} + \)\(93\!\cdots\!29\)\( T^{3} + \)\(57\!\cdots\!56\)\( T^{4} + \)\(18\!\cdots\!87\)\( T^{5} + \)\(65\!\cdots\!05\)\( T^{6} + \)\(26\!\cdots\!66\)\( p T^{7} + \)\(65\!\cdots\!05\)\( p^{8} T^{8} + \)\(18\!\cdots\!87\)\( p^{16} T^{9} + \)\(57\!\cdots\!56\)\( p^{24} T^{10} + \)\(93\!\cdots\!29\)\( p^{32} T^{11} + 3410148934050418 p^{40} T^{12} + 18360841 p^{48} T^{13} + p^{56} T^{14} )^{2} \) | |
| 79 | \( 1 + 32771822 T - 3895511802864963 T^{2} - \)\(26\!\cdots\!78\)\( T^{3} + \)\(38\!\cdots\!69\)\( T^{4} + \)\(75\!\cdots\!76\)\( T^{5} + \)\(18\!\cdots\!40\)\( T^{6} - \)\(84\!\cdots\!68\)\( T^{7} - \)\(64\!\cdots\!56\)\( T^{8} - \)\(10\!\cdots\!72\)\( T^{9} + \)\(70\!\cdots\!36\)\( T^{10} + \)\(42\!\cdots\!52\)\( T^{11} + \)\(63\!\cdots\!70\)\( T^{12} - \)\(36\!\cdots\!68\)\( T^{13} - \)\(23\!\cdots\!86\)\( T^{14} - \)\(36\!\cdots\!68\)\( p^{8} T^{15} + \)\(63\!\cdots\!70\)\( p^{16} T^{16} + \)\(42\!\cdots\!52\)\( p^{24} T^{17} + \)\(70\!\cdots\!36\)\( p^{32} T^{18} - \)\(10\!\cdots\!72\)\( p^{40} T^{19} - \)\(64\!\cdots\!56\)\( p^{48} T^{20} - \)\(84\!\cdots\!68\)\( p^{56} T^{21} + \)\(18\!\cdots\!40\)\( p^{64} T^{22} + \)\(75\!\cdots\!76\)\( p^{72} T^{23} + \)\(38\!\cdots\!69\)\( p^{80} T^{24} - \)\(26\!\cdots\!78\)\( p^{88} T^{25} - 3895511802864963 p^{96} T^{26} + 32771822 p^{104} T^{27} + p^{112} T^{28} \) | |
| 83 | \( 1 + 198915996 T + 358046298144425 p T^{2} + \)\(32\!\cdots\!88\)\( T^{3} + \)\(30\!\cdots\!53\)\( T^{4} + \)\(25\!\cdots\!80\)\( T^{5} + \)\(18\!\cdots\!52\)\( T^{6} + \)\(12\!\cdots\!08\)\( T^{7} + \)\(79\!\cdots\!60\)\( T^{8} + \)\(57\!\cdots\!84\)\( p T^{9} + \)\(27\!\cdots\!48\)\( T^{10} + \)\(14\!\cdots\!80\)\( T^{11} + \)\(94\!\cdots\!02\)\( p T^{12} + \)\(39\!\cdots\!16\)\( T^{13} + \)\(19\!\cdots\!30\)\( T^{14} + \)\(39\!\cdots\!16\)\( p^{8} T^{15} + \)\(94\!\cdots\!02\)\( p^{17} T^{16} + \)\(14\!\cdots\!80\)\( p^{24} T^{17} + \)\(27\!\cdots\!48\)\( p^{32} T^{18} + \)\(57\!\cdots\!84\)\( p^{41} T^{19} + \)\(79\!\cdots\!60\)\( p^{48} T^{20} + \)\(12\!\cdots\!08\)\( p^{56} T^{21} + \)\(18\!\cdots\!52\)\( p^{64} T^{22} + \)\(25\!\cdots\!80\)\( p^{72} T^{23} + \)\(30\!\cdots\!53\)\( p^{80} T^{24} + \)\(32\!\cdots\!88\)\( p^{88} T^{25} + 358046298144425 p^{97} T^{26} + 198915996 p^{104} T^{27} + p^{112} T^{28} \) | |
| 89 | \( 1 - 41326530938392262 T^{2} + \)\(81\!\cdots\!39\)\( T^{4} - \)\(10\!\cdots\!84\)\( T^{6} + \)\(94\!\cdots\!49\)\( T^{8} - \)\(65\!\cdots\!62\)\( T^{10} + \)\(35\!\cdots\!79\)\( T^{12} - \)\(15\!\cdots\!80\)\( T^{14} + \)\(35\!\cdots\!79\)\( p^{16} T^{16} - \)\(65\!\cdots\!62\)\( p^{32} T^{18} + \)\(94\!\cdots\!49\)\( p^{48} T^{20} - \)\(10\!\cdots\!84\)\( p^{64} T^{22} + \)\(81\!\cdots\!39\)\( p^{80} T^{24} - 41326530938392262 p^{96} T^{26} + p^{112} T^{28} \) | |
| 97 | \( 1 - 127049161 T - 22022268725944599 T^{2} + \)\(30\!\cdots\!82\)\( T^{3} + \)\(27\!\cdots\!36\)\( T^{4} - \)\(40\!\cdots\!60\)\( T^{5} - \)\(25\!\cdots\!63\)\( T^{6} + \)\(47\!\cdots\!49\)\( T^{7} + \)\(11\!\cdots\!76\)\( T^{8} - \)\(41\!\cdots\!45\)\( T^{9} + \)\(19\!\cdots\!07\)\( T^{10} + \)\(27\!\cdots\!20\)\( p T^{11} - \)\(76\!\cdots\!13\)\( T^{12} - \)\(85\!\cdots\!37\)\( T^{13} + \)\(82\!\cdots\!90\)\( T^{14} - \)\(85\!\cdots\!37\)\( p^{8} T^{15} - \)\(76\!\cdots\!13\)\( p^{16} T^{16} + \)\(27\!\cdots\!20\)\( p^{25} T^{17} + \)\(19\!\cdots\!07\)\( p^{32} T^{18} - \)\(41\!\cdots\!45\)\( p^{40} T^{19} + \)\(11\!\cdots\!76\)\( p^{48} T^{20} + \)\(47\!\cdots\!49\)\( p^{56} T^{21} - \)\(25\!\cdots\!63\)\( p^{64} T^{22} - \)\(40\!\cdots\!60\)\( p^{72} T^{23} + \)\(27\!\cdots\!36\)\( p^{80} T^{24} + \)\(30\!\cdots\!82\)\( p^{88} T^{25} - 22022268725944599 p^{96} T^{26} - 127049161 p^{104} T^{27} + p^{112} T^{28} \) | |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{28} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−6.04845802692798669774395671846, −5.99259420850941214303848521549, −5.67150760744565750228719329806, −5.56153928530669329441785329668, −5.52847757321289050167741889985, −5.19065848468107296842545980032, −4.84297270831759192925185483629, −4.78980994530513214802469731579, −4.43767167242911547690337494440, −4.25711382585556720638867140868, −4.21489295913894054357223656485, −4.17227290223665537786876960297, −4.11866726565879182457521211816, −3.50079334468490511309708086589, −3.26134991249582470067155627626, −2.85433045701350068535550668820, −2.62538156552917039955036203136, −2.21777943348888205037041890160, −2.04319763270510197248861529935, −1.69526611868712884942989270072, −1.50946181416601758716779240250, −1.16720721083289990081338060380, −0.53681807599824510589392014368, −0.30478727359902946164962514141, −0.23684077861348057082174797875, 0.23684077861348057082174797875, 0.30478727359902946164962514141, 0.53681807599824510589392014368, 1.16720721083289990081338060380, 1.50946181416601758716779240250, 1.69526611868712884942989270072, 2.04319763270510197248861529935, 2.21777943348888205037041890160, 2.62538156552917039955036203136, 2.85433045701350068535550668820, 3.26134991249582470067155627626, 3.50079334468490511309708086589, 4.11866726565879182457521211816, 4.17227290223665537786876960297, 4.21489295913894054357223656485, 4.25711382585556720638867140868, 4.43767167242911547690337494440, 4.78980994530513214802469731579, 4.84297270831759192925185483629, 5.19065848468107296842545980032, 5.52847757321289050167741889985, 5.56153928530669329441785329668, 5.67150760744565750228719329806, 5.99259420850941214303848521549, 6.04845802692798669774395671846
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.