Dirichlet series
| L(s) = 1 | − 4·2-s − 16·4-s − 6·7-s + 81·8-s − 29·11-s − 24·13-s + 24·14-s + 76·16-s − 79·17-s − 75·19-s + 116·22-s − 318·23-s + 96·26-s + 96·28-s − 106·29-s + 60·31-s − 754·32-s + 316·34-s + 84·37-s + 300·38-s + 353·41-s + 426·43-s + 464·44-s + 1.27e3·46-s − 1.21e3·47-s − 1.70e3·49-s + 384·52-s + ⋯ |
| L(s) = 1 | − 1.41·2-s − 2·4-s − 0.323·7-s + 3.57·8-s − 0.794·11-s − 0.512·13-s + 0.458·14-s + 1.18·16-s − 1.12·17-s − 0.905·19-s + 1.12·22-s − 2.88·23-s + 0.724·26-s + 0.647·28-s − 0.678·29-s + 0.347·31-s − 4.16·32-s + 1.59·34-s + 0.373·37-s + 1.28·38-s + 1.34·41-s + 1.51·43-s + 1.58·44-s + 4.07·46-s − 3.75·47-s − 4.97·49-s + 1.02·52-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{48} \cdot 5^{24}\right)^{s/2} \, \Gamma_{\C}(s)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{48} \cdot 5^{24}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(24\) |
| Conductor: | \(3^{48} \cdot 5^{24}\) |
| Sign: | $1$ |
| Analytic conductor: | \(8.46239\times 10^{24}\) |
| Root analytic conductor: | \(10.9306\) |
| Motivic weight: | \(3\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(12\) |
| Selberg data: | \((24,\ 3^{48} \cdot 5^{24} ,\ ( \ : [3/2]^{12} ),\ 1 )\) |
Particular Values
| \(L(2)\) | \(=\) | \(0\) |
| \(L(\frac12)\) | \(=\) | \(0\) |
| \(L(\frac{5}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 \) | |
| good | 2 | \( 1 + p^{2} T + p^{5} T^{2} + 111 T^{3} + 139 p^{2} T^{4} + 929 p T^{5} + 3707 p T^{6} + 24689 T^{7} + 85981 T^{8} + 135621 p T^{9} + 52607 p^{4} T^{10} + 154697 p^{4} T^{11} + 446555 p^{4} T^{12} + 154697 p^{7} T^{13} + 52607 p^{10} T^{14} + 135621 p^{10} T^{15} + 85981 p^{12} T^{16} + 24689 p^{15} T^{17} + 3707 p^{19} T^{18} + 929 p^{22} T^{19} + 139 p^{26} T^{20} + 111 p^{27} T^{21} + p^{35} T^{22} + p^{35} T^{23} + p^{36} T^{24} \) |
| 7 | \( 1 + 6 T + 249 p T^{2} + 7724 T^{3} + 1715712 T^{4} + 5730846 T^{5} + 171598615 p T^{6} + 2866389282 T^{7} + 657865636002 T^{8} + 1117701527010 T^{9} + 293934672252558 T^{10} + 384876917744652 T^{11} + 109871398007648142 T^{12} + 384876917744652 p^{3} T^{13} + 293934672252558 p^{6} T^{14} + 1117701527010 p^{9} T^{15} + 657865636002 p^{12} T^{16} + 2866389282 p^{15} T^{17} + 171598615 p^{19} T^{18} + 5730846 p^{21} T^{19} + 1715712 p^{24} T^{20} + 7724 p^{27} T^{21} + 249 p^{31} T^{22} + 6 p^{33} T^{23} + p^{36} T^{24} \) | |
| 11 | \( 1 + 29 T + 7487 T^{2} + 21414 p T^{3} + 29122939 T^{4} + 898403861 T^{5} + 76825771222 T^{6} + 197977414289 p T^{7} + 153195352720180 T^{8} + 3902099891534001 T^{9} + 249211874124216143 T^{10} + 5786548817819867374 T^{11} + \)\(35\!\cdots\!40\)\( T^{12} + 5786548817819867374 p^{3} T^{13} + 249211874124216143 p^{6} T^{14} + 3902099891534001 p^{9} T^{15} + 153195352720180 p^{12} T^{16} + 197977414289 p^{16} T^{17} + 76825771222 p^{18} T^{18} + 898403861 p^{21} T^{19} + 29122939 p^{24} T^{20} + 21414 p^{28} T^{21} + 7487 p^{30} T^{22} + 29 p^{33} T^{23} + p^{36} T^{24} \) | |
| 13 | \( 1 + 24 T + 13830 T^{2} + 591566 T^{3} + 97882764 T^{4} + 418801686 p T^{5} + 496854797755 T^{6} + 29119158539046 T^{7} + 1978304229010476 T^{8} + 108709107019106998 T^{9} + 6202118140708859745 T^{10} + \)\(30\!\cdots\!28\)\( T^{11} + \)\(11\!\cdots\!66\)\( p T^{12} + \)\(30\!\cdots\!28\)\( p^{3} T^{13} + 6202118140708859745 p^{6} T^{14} + 108709107019106998 p^{9} T^{15} + 1978304229010476 p^{12} T^{16} + 29119158539046 p^{15} T^{17} + 496854797755 p^{18} T^{18} + 418801686 p^{22} T^{19} + 97882764 p^{24} T^{20} + 591566 p^{27} T^{21} + 13830 p^{30} T^{22} + 24 p^{33} T^{23} + p^{36} T^{24} \) | |
| 17 | \( 1 + 79 T + 34793 T^{2} + 2020506 T^{3} + 564439972 T^{4} + 23450427034 T^{5} + 5771918743639 T^{6} + 155921314090247 T^{7} + 42595334197320988 T^{8} + 621343164446979207 T^{9} + \)\(25\!\cdots\!78\)\( T^{10} + \)\(16\!\cdots\!67\)\( T^{11} + \)\(12\!\cdots\!78\)\( T^{12} + \)\(16\!\cdots\!67\)\( p^{3} T^{13} + \)\(25\!\cdots\!78\)\( p^{6} T^{14} + 621343164446979207 p^{9} T^{15} + 42595334197320988 p^{12} T^{16} + 155921314090247 p^{15} T^{17} + 5771918743639 p^{18} T^{18} + 23450427034 p^{21} T^{19} + 564439972 p^{24} T^{20} + 2020506 p^{27} T^{21} + 34793 p^{30} T^{22} + 79 p^{33} T^{23} + p^{36} T^{24} \) | |
| 19 | \( 1 + 75 T + 34881 T^{2} + 2656426 T^{3} + 636553851 T^{4} + 45462845721 T^{5} + 8008663071802 T^{6} + 27447120428373 p T^{7} + 78188286606942186 T^{8} + 4575880322276863001 T^{9} + \)\(63\!\cdots\!16\)\( T^{10} + \)\(34\!\cdots\!31\)\( T^{11} + \)\(45\!\cdots\!06\)\( T^{12} + \)\(34\!\cdots\!31\)\( p^{3} T^{13} + \)\(63\!\cdots\!16\)\( p^{6} T^{14} + 4575880322276863001 p^{9} T^{15} + 78188286606942186 p^{12} T^{16} + 27447120428373 p^{16} T^{17} + 8008663071802 p^{18} T^{18} + 45462845721 p^{21} T^{19} + 636553851 p^{24} T^{20} + 2656426 p^{27} T^{21} + 34881 p^{30} T^{22} + 75 p^{33} T^{23} + p^{36} T^{24} \) | |
| 23 | \( 1 + 318 T + 120489 T^{2} + 25777992 T^{3} + 5935350585 T^{4} + 998197560480 T^{5} + 176554594406210 T^{6} + 25021734059582460 T^{7} + 3717036685461918618 T^{8} + \)\(46\!\cdots\!76\)\( T^{9} + \)\(60\!\cdots\!39\)\( T^{10} + \)\(68\!\cdots\!86\)\( T^{11} + \)\(80\!\cdots\!56\)\( T^{12} + \)\(68\!\cdots\!86\)\( p^{3} T^{13} + \)\(60\!\cdots\!39\)\( p^{6} T^{14} + \)\(46\!\cdots\!76\)\( p^{9} T^{15} + 3717036685461918618 p^{12} T^{16} + 25021734059582460 p^{15} T^{17} + 176554594406210 p^{18} T^{18} + 998197560480 p^{21} T^{19} + 5935350585 p^{24} T^{20} + 25777992 p^{27} T^{21} + 120489 p^{30} T^{22} + 318 p^{33} T^{23} + p^{36} T^{24} \) | |
| 29 | \( 1 + 106 T + 135611 T^{2} + 7637010 T^{3} + 9064192468 T^{4} + 115444489612 T^{5} + 392590673756995 T^{6} - 14926383459249220 T^{7} + 12339728787268393792 T^{8} - \)\(11\!\cdots\!06\)\( T^{9} + \)\(31\!\cdots\!72\)\( T^{10} - \)\(45\!\cdots\!86\)\( T^{11} + \)\(76\!\cdots\!86\)\( T^{12} - \)\(45\!\cdots\!86\)\( p^{3} T^{13} + \)\(31\!\cdots\!72\)\( p^{6} T^{14} - \)\(11\!\cdots\!06\)\( p^{9} T^{15} + 12339728787268393792 p^{12} T^{16} - 14926383459249220 p^{15} T^{17} + 392590673756995 p^{18} T^{18} + 115444489612 p^{21} T^{19} + 9064192468 p^{24} T^{20} + 7637010 p^{27} T^{21} + 135611 p^{30} T^{22} + 106 p^{33} T^{23} + p^{36} T^{24} \) | |
| 31 | \( 1 - 60 T + 174210 T^{2} - 20665748 T^{3} + 14593122204 T^{4} - 2515124122206 T^{5} + 834961756515925 T^{6} - 171611841033878742 T^{7} + 38170316406808693566 T^{8} - \)\(80\!\cdots\!92\)\( T^{9} + \)\(14\!\cdots\!43\)\( T^{10} - \)\(29\!\cdots\!24\)\( T^{11} + \)\(47\!\cdots\!54\)\( T^{12} - \)\(29\!\cdots\!24\)\( p^{3} T^{13} + \)\(14\!\cdots\!43\)\( p^{6} T^{14} - \)\(80\!\cdots\!92\)\( p^{9} T^{15} + 38170316406808693566 p^{12} T^{16} - 171611841033878742 p^{15} T^{17} + 834961756515925 p^{18} T^{18} - 2515124122206 p^{21} T^{19} + 14593122204 p^{24} T^{20} - 20665748 p^{27} T^{21} + 174210 p^{30} T^{22} - 60 p^{33} T^{23} + p^{36} T^{24} \) | |
| 37 | \( 1 - 84 T + 315861 T^{2} - 23606110 T^{3} + 53180717667 T^{4} - 3694526636376 T^{5} + 6133086231906502 T^{6} - 396332408056958910 T^{7} + \)\(53\!\cdots\!16\)\( T^{8} - \)\(32\!\cdots\!16\)\( T^{9} + \)\(36\!\cdots\!59\)\( T^{10} - \)\(20\!\cdots\!88\)\( T^{11} + \)\(20\!\cdots\!92\)\( T^{12} - \)\(20\!\cdots\!88\)\( p^{3} T^{13} + \)\(36\!\cdots\!59\)\( p^{6} T^{14} - \)\(32\!\cdots\!16\)\( p^{9} T^{15} + \)\(53\!\cdots\!16\)\( p^{12} T^{16} - 396332408056958910 p^{15} T^{17} + 6133086231906502 p^{18} T^{18} - 3694526636376 p^{21} T^{19} + 53180717667 p^{24} T^{20} - 23606110 p^{27} T^{21} + 315861 p^{30} T^{22} - 84 p^{33} T^{23} + p^{36} T^{24} \) | |
| 41 | \( 1 - 353 T + 436886 T^{2} - 107057439 T^{3} + 86925906826 T^{4} - 15827041462469 T^{5} + 11349176196270904 T^{6} - 1567240192507505671 T^{7} + \)\(11\!\cdots\!99\)\( T^{8} - \)\(12\!\cdots\!00\)\( T^{9} + \)\(94\!\cdots\!55\)\( T^{10} - \)\(83\!\cdots\!17\)\( T^{11} + \)\(68\!\cdots\!68\)\( T^{12} - \)\(83\!\cdots\!17\)\( p^{3} T^{13} + \)\(94\!\cdots\!55\)\( p^{6} T^{14} - \)\(12\!\cdots\!00\)\( p^{9} T^{15} + \)\(11\!\cdots\!99\)\( p^{12} T^{16} - 1567240192507505671 p^{15} T^{17} + 11349176196270904 p^{18} T^{18} - 15827041462469 p^{21} T^{19} + 86925906826 p^{24} T^{20} - 107057439 p^{27} T^{21} + 436886 p^{30} T^{22} - 353 p^{33} T^{23} + p^{36} T^{24} \) | |
| 43 | \( 1 - 426 T + 395214 T^{2} - 112459144 T^{3} + 64539424653 T^{4} - 13751338463220 T^{5} + 6961683595859626 T^{6} - 1248427503010451790 T^{7} + \)\(66\!\cdots\!34\)\( T^{8} - \)\(11\!\cdots\!42\)\( T^{9} + \)\(63\!\cdots\!54\)\( T^{10} - \)\(10\!\cdots\!52\)\( T^{11} + \)\(54\!\cdots\!65\)\( T^{12} - \)\(10\!\cdots\!52\)\( p^{3} T^{13} + \)\(63\!\cdots\!54\)\( p^{6} T^{14} - \)\(11\!\cdots\!42\)\( p^{9} T^{15} + \)\(66\!\cdots\!34\)\( p^{12} T^{16} - 1248427503010451790 p^{15} T^{17} + 6961683595859626 p^{18} T^{18} - 13751338463220 p^{21} T^{19} + 64539424653 p^{24} T^{20} - 112459144 p^{27} T^{21} + 395214 p^{30} T^{22} - 426 p^{33} T^{23} + p^{36} T^{24} \) | |
| 47 | \( 1 + 1210 T + 1331672 T^{2} + 932288892 T^{3} + 613104764113 T^{4} + 316151901078196 T^{5} + 158839612717586482 T^{6} + 67793687952816840644 T^{7} + \)\(29\!\cdots\!86\)\( T^{8} + \)\(10\!\cdots\!92\)\( T^{9} + \)\(41\!\cdots\!02\)\( T^{10} + \)\(14\!\cdots\!86\)\( T^{11} + \)\(48\!\cdots\!92\)\( T^{12} + \)\(14\!\cdots\!86\)\( p^{3} T^{13} + \)\(41\!\cdots\!02\)\( p^{6} T^{14} + \)\(10\!\cdots\!92\)\( p^{9} T^{15} + \)\(29\!\cdots\!86\)\( p^{12} T^{16} + 67793687952816840644 p^{15} T^{17} + 158839612717586482 p^{18} T^{18} + 316151901078196 p^{21} T^{19} + 613104764113 p^{24} T^{20} + 932288892 p^{27} T^{21} + 1331672 p^{30} T^{22} + 1210 p^{33} T^{23} + p^{36} T^{24} \) | |
| 53 | \( 1 + 448 T + 803618 T^{2} + 438573258 T^{3} + 365066600488 T^{4} + 195422249670982 T^{5} + 121036091477151187 T^{6} + 57297209313055497578 T^{7} + \)\(30\!\cdots\!36\)\( T^{8} + \)\(12\!\cdots\!10\)\( T^{9} + \)\(60\!\cdots\!09\)\( T^{10} + \)\(23\!\cdots\!84\)\( T^{11} + \)\(99\!\cdots\!50\)\( T^{12} + \)\(23\!\cdots\!84\)\( p^{3} T^{13} + \)\(60\!\cdots\!09\)\( p^{6} T^{14} + \)\(12\!\cdots\!10\)\( p^{9} T^{15} + \)\(30\!\cdots\!36\)\( p^{12} T^{16} + 57297209313055497578 p^{15} T^{17} + 121036091477151187 p^{18} T^{18} + 195422249670982 p^{21} T^{19} + 365066600488 p^{24} T^{20} + 438573258 p^{27} T^{21} + 803618 p^{30} T^{22} + 448 p^{33} T^{23} + p^{36} T^{24} \) | |
| 59 | \( 1 + 482 T + 1405445 T^{2} + 652008960 T^{3} + 1033308269617 T^{4} + 448870499318174 T^{5} + 514722592337198512 T^{6} + \)\(20\!\cdots\!44\)\( T^{7} + \)\(19\!\cdots\!26\)\( T^{8} + \)\(70\!\cdots\!22\)\( T^{9} + \)\(55\!\cdots\!39\)\( T^{10} + \)\(18\!\cdots\!72\)\( T^{11} + \)\(21\!\cdots\!39\)\( p T^{12} + \)\(18\!\cdots\!72\)\( p^{3} T^{13} + \)\(55\!\cdots\!39\)\( p^{6} T^{14} + \)\(70\!\cdots\!22\)\( p^{9} T^{15} + \)\(19\!\cdots\!26\)\( p^{12} T^{16} + \)\(20\!\cdots\!44\)\( p^{15} T^{17} + 514722592337198512 p^{18} T^{18} + 448870499318174 p^{21} T^{19} + 1033308269617 p^{24} T^{20} + 652008960 p^{27} T^{21} + 1405445 p^{30} T^{22} + 482 p^{33} T^{23} + p^{36} T^{24} \) | |
| 61 | \( 1 - 402 T + 1589847 T^{2} - 492088838 T^{3} + 1231554041277 T^{4} - 282692249695446 T^{5} + 617065409547426046 T^{6} - 97991579888387135676 T^{7} + \)\(22\!\cdots\!58\)\( T^{8} - \)\(23\!\cdots\!18\)\( T^{9} + \)\(65\!\cdots\!51\)\( T^{10} - \)\(44\!\cdots\!64\)\( T^{11} + \)\(16\!\cdots\!08\)\( T^{12} - \)\(44\!\cdots\!64\)\( p^{3} T^{13} + \)\(65\!\cdots\!51\)\( p^{6} T^{14} - \)\(23\!\cdots\!18\)\( p^{9} T^{15} + \)\(22\!\cdots\!58\)\( p^{12} T^{16} - 97991579888387135676 p^{15} T^{17} + 617065409547426046 p^{18} T^{18} - 282692249695446 p^{21} T^{19} + 1231554041277 p^{24} T^{20} - 492088838 p^{27} T^{21} + 1589847 p^{30} T^{22} - 402 p^{33} T^{23} + p^{36} T^{24} \) | |
| 67 | \( 1 - 3 p T + 1775580 T^{2} - 105472735 T^{3} + 1554197215461 T^{4} + 70344746855304 T^{5} + 935863675464881488 T^{6} + \)\(10\!\cdots\!66\)\( T^{7} + \)\(44\!\cdots\!68\)\( T^{8} + \)\(10\!\cdots\!56\)\( p T^{9} + \)\(17\!\cdots\!45\)\( T^{10} + \)\(27\!\cdots\!11\)\( T^{11} + \)\(56\!\cdots\!02\)\( T^{12} + \)\(27\!\cdots\!11\)\( p^{3} T^{13} + \)\(17\!\cdots\!45\)\( p^{6} T^{14} + \)\(10\!\cdots\!56\)\( p^{10} T^{15} + \)\(44\!\cdots\!68\)\( p^{12} T^{16} + \)\(10\!\cdots\!66\)\( p^{15} T^{17} + 935863675464881488 p^{18} T^{18} + 70344746855304 p^{21} T^{19} + 1554197215461 p^{24} T^{20} - 105472735 p^{27} T^{21} + 1775580 p^{30} T^{22} - 3 p^{34} T^{23} + p^{36} T^{24} \) | |
| 71 | \( 1 - 944 T + 1981262 T^{2} - 964318104 T^{3} + 1243525132846 T^{4} - 297943571263940 T^{5} + 529095935741066197 T^{6} - \)\(20\!\cdots\!82\)\( T^{7} + \)\(30\!\cdots\!82\)\( T^{8} - \)\(15\!\cdots\!10\)\( T^{9} + \)\(11\!\cdots\!23\)\( T^{10} - \)\(49\!\cdots\!08\)\( T^{11} + \)\(31\!\cdots\!62\)\( T^{12} - \)\(49\!\cdots\!08\)\( p^{3} T^{13} + \)\(11\!\cdots\!23\)\( p^{6} T^{14} - \)\(15\!\cdots\!10\)\( p^{9} T^{15} + \)\(30\!\cdots\!82\)\( p^{12} T^{16} - \)\(20\!\cdots\!82\)\( p^{15} T^{17} + 529095935741066197 p^{18} T^{18} - 297943571263940 p^{21} T^{19} + 1243525132846 p^{24} T^{20} - 964318104 p^{27} T^{21} + 1981262 p^{30} T^{22} - 944 p^{33} T^{23} + p^{36} T^{24} \) | |
| 73 | \( 1 - 453 T + 2235741 T^{2} - 864799528 T^{3} + 2515203790710 T^{4} - 816973398713424 T^{5} + 1927445586169077778 T^{6} - \)\(51\!\cdots\!18\)\( T^{7} + \)\(11\!\cdots\!33\)\( T^{8} - \)\(24\!\cdots\!81\)\( T^{9} + \)\(54\!\cdots\!86\)\( T^{10} - \)\(99\!\cdots\!75\)\( T^{11} + \)\(22\!\cdots\!92\)\( T^{12} - \)\(99\!\cdots\!75\)\( p^{3} T^{13} + \)\(54\!\cdots\!86\)\( p^{6} T^{14} - \)\(24\!\cdots\!81\)\( p^{9} T^{15} + \)\(11\!\cdots\!33\)\( p^{12} T^{16} - \)\(51\!\cdots\!18\)\( p^{15} T^{17} + 1927445586169077778 p^{18} T^{18} - 816973398713424 p^{21} T^{19} + 2515203790710 p^{24} T^{20} - 864799528 p^{27} T^{21} + 2235741 p^{30} T^{22} - 453 p^{33} T^{23} + p^{36} T^{24} \) | |
| 79 | \( 1 - 258 T + 2573061 T^{2} - 304848812 T^{3} + 3367554071019 T^{4} + 109613061795720 T^{5} + 2895049289666111044 T^{6} + \)\(61\!\cdots\!98\)\( T^{7} + \)\(18\!\cdots\!88\)\( T^{8} + \)\(76\!\cdots\!38\)\( T^{9} + \)\(97\!\cdots\!15\)\( T^{10} + \)\(56\!\cdots\!30\)\( T^{11} + \)\(48\!\cdots\!28\)\( T^{12} + \)\(56\!\cdots\!30\)\( p^{3} T^{13} + \)\(97\!\cdots\!15\)\( p^{6} T^{14} + \)\(76\!\cdots\!38\)\( p^{9} T^{15} + \)\(18\!\cdots\!88\)\( p^{12} T^{16} + \)\(61\!\cdots\!98\)\( p^{15} T^{17} + 2895049289666111044 p^{18} T^{18} + 109613061795720 p^{21} T^{19} + 3367554071019 p^{24} T^{20} - 304848812 p^{27} T^{21} + 2573061 p^{30} T^{22} - 258 p^{33} T^{23} + p^{36} T^{24} \) | |
| 83 | \( 1 + 3012 T + 8506626 T^{2} + 16172506950 T^{3} + 28066163447937 T^{4} + 40181001205607574 T^{5} + 639031614231598207 p T^{6} + \)\(61\!\cdots\!16\)\( T^{7} + \)\(66\!\cdots\!84\)\( T^{8} + \)\(65\!\cdots\!86\)\( T^{9} + \)\(59\!\cdots\!86\)\( T^{10} + \)\(50\!\cdots\!02\)\( T^{11} + \)\(39\!\cdots\!29\)\( T^{12} + \)\(50\!\cdots\!02\)\( p^{3} T^{13} + \)\(59\!\cdots\!86\)\( p^{6} T^{14} + \)\(65\!\cdots\!86\)\( p^{9} T^{15} + \)\(66\!\cdots\!84\)\( p^{12} T^{16} + \)\(61\!\cdots\!16\)\( p^{15} T^{17} + 639031614231598207 p^{19} T^{18} + 40181001205607574 p^{21} T^{19} + 28066163447937 p^{24} T^{20} + 16172506950 p^{27} T^{21} + 8506626 p^{30} T^{22} + 3012 p^{33} T^{23} + p^{36} T^{24} \) | |
| 89 | \( 1 - 738 T + 6271305 T^{2} - 3292949034 T^{3} + 17452341965856 T^{4} - 5690926315510776 T^{5} + 28928071788736123913 T^{6} - \)\(38\!\cdots\!42\)\( T^{7} + \)\(32\!\cdots\!87\)\( T^{8} + \)\(15\!\cdots\!24\)\( T^{9} + \)\(28\!\cdots\!03\)\( T^{10} + \)\(47\!\cdots\!84\)\( T^{11} + \)\(21\!\cdots\!35\)\( T^{12} + \)\(47\!\cdots\!84\)\( p^{3} T^{13} + \)\(28\!\cdots\!03\)\( p^{6} T^{14} + \)\(15\!\cdots\!24\)\( p^{9} T^{15} + \)\(32\!\cdots\!87\)\( p^{12} T^{16} - \)\(38\!\cdots\!42\)\( p^{15} T^{17} + 28928071788736123913 p^{18} T^{18} - 5690926315510776 p^{21} T^{19} + 17452341965856 p^{24} T^{20} - 3292949034 p^{27} T^{21} + 6271305 p^{30} T^{22} - 738 p^{33} T^{23} + p^{36} T^{24} \) | |
| 97 | \( 1 - 318 T + 5144037 T^{2} - 1666625662 T^{3} + 12642592386048 T^{4} - 4955552245617204 T^{5} + 21381142564847396161 T^{6} - \)\(10\!\cdots\!14\)\( T^{7} + \)\(29\!\cdots\!75\)\( T^{8} - \)\(14\!\cdots\!80\)\( T^{9} + \)\(34\!\cdots\!47\)\( T^{10} - \)\(15\!\cdots\!96\)\( T^{11} + \)\(34\!\cdots\!51\)\( T^{12} - \)\(15\!\cdots\!96\)\( p^{3} T^{13} + \)\(34\!\cdots\!47\)\( p^{6} T^{14} - \)\(14\!\cdots\!80\)\( p^{9} T^{15} + \)\(29\!\cdots\!75\)\( p^{12} T^{16} - \)\(10\!\cdots\!14\)\( p^{15} T^{17} + 21381142564847396161 p^{18} T^{18} - 4955552245617204 p^{21} T^{19} + 12642592386048 p^{24} T^{20} - 1666625662 p^{27} T^{21} + 5144037 p^{30} T^{22} - 318 p^{33} T^{23} + p^{36} T^{24} \) | |
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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
−2.87368719784667421652227398912, −2.78303315084328892193672160785, −2.74262506031920813705873828158, −2.60870054024066104567193722063, −2.56754324659965458075789413783, −2.42904086226960490566849376904, −2.39708956736082055102104184262, −2.38657425812594927714958365594, −2.17704435170297860396787945654, −2.14875566078381554863629995520, −2.05229264480280655882022329726, −1.99088967485610600731475839076, −1.89285836251520250132303775294, −1.88270002536237362272727752626, −1.63128983579297384769371177964, −1.49745711978674706406903274114, −1.28948315761324558777951470872, −1.26074859553949280278086431834, −1.22741049025787467493528134708, −1.16352610528518942797472868814, −1.11395798392955216517718949228, −1.03581819255345657279803739908, −0.935123661919987389113527475467, −0.865986817303080596749848257918, −0.864771407927055956235531452416, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0.864771407927055956235531452416, 0.865986817303080596749848257918, 0.935123661919987389113527475467, 1.03581819255345657279803739908, 1.11395798392955216517718949228, 1.16352610528518942797472868814, 1.22741049025787467493528134708, 1.26074859553949280278086431834, 1.28948315761324558777951470872, 1.49745711978674706406903274114, 1.63128983579297384769371177964, 1.88270002536237362272727752626, 1.89285836251520250132303775294, 1.99088967485610600731475839076, 2.05229264480280655882022329726, 2.14875566078381554863629995520, 2.17704435170297860396787945654, 2.38657425812594927714958365594, 2.39708956736082055102104184262, 2.42904086226960490566849376904, 2.56754324659965458075789413783, 2.60870054024066104567193722063, 2.74262506031920813705873828158, 2.78303315084328892193672160785, 2.87368719784667421652227398912
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.