Dirichlet series
| L(s) = 1 | − 14·3-s + 10·5-s + 44·7-s + 9-s − 46·11-s + 34·13-s − 140·15-s + 36·17-s − 148·19-s − 616·21-s + 328·23-s − 457·25-s + 918·27-s − 348·29-s + 374·31-s + 644·33-s + 440·35-s + 340·37-s − 476·39-s + 32·41-s − 462·43-s + 10·45-s + 434·47-s − 336·49-s − 504·51-s − 610·53-s − 460·55-s + ⋯ |
| L(s) = 1 | − 2.69·3-s + 0.894·5-s + 2.37·7-s + 1/27·9-s − 1.26·11-s + 0.725·13-s − 2.40·15-s + 0.513·17-s − 1.78·19-s − 6.40·21-s + 2.97·23-s − 3.65·25-s + 6.54·27-s − 2.22·29-s + 2.16·31-s + 3.39·33-s + 2.12·35-s + 1.51·37-s − 1.95·39-s + 0.121·41-s − 1.63·43-s + 0.0331·45-s + 1.34·47-s − 0.979·49-s − 1.38·51-s − 1.58·53-s − 1.12·55-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{72} \cdot 29^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{72} \cdot 29^{12}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(24\) |
| Conductor: | \(2^{72} \cdot 29^{12}\) |
| Sign: | $1$ |
| Analytic conductor: | \(2.97391\times 10^{24}\) |
| Root analytic conductor: | \(10.4645\) |
| Motivic weight: | \(3\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((24,\ 2^{72} \cdot 29^{12} ,\ ( \ : [3/2]^{12} ),\ 1 )\) |
Particular Values
| \(L(2)\) | \(\approx\) | \(5.705187975\) |
| \(L(\frac12)\) | \(\approx\) | \(5.705187975\) |
| \(L(\frac{5}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 2 | \( 1 \) |
| 29 | \( ( 1 + p T )^{12} \) | |
| good | 3 | \( 1 + 14 T + 65 p T^{2} + 1798 T^{3} + 15686 T^{4} + 110086 T^{5} + 730397 T^{6} + 4150958 T^{7} + 22566016 T^{8} + 35930074 p T^{9} + 522075287 T^{10} + 260479334 p^{2} T^{11} + 12180883156 T^{12} + 260479334 p^{5} T^{13} + 522075287 p^{6} T^{14} + 35930074 p^{10} T^{15} + 22566016 p^{12} T^{16} + 4150958 p^{15} T^{17} + 730397 p^{18} T^{18} + 110086 p^{21} T^{19} + 15686 p^{24} T^{20} + 1798 p^{27} T^{21} + 65 p^{31} T^{22} + 14 p^{33} T^{23} + p^{36} T^{24} \) |
| 5 | \( 1 - 2 p T + 557 T^{2} - 7098 T^{3} + 183722 T^{4} - 2482934 T^{5} + 43410603 T^{6} - 569874862 T^{7} + 8284116752 T^{8} - 98048828554 T^{9} + 263270836661 p T^{10} - 2784444761538 p T^{11} + 7144023302312 p^{2} T^{12} - 2784444761538 p^{4} T^{13} + 263270836661 p^{7} T^{14} - 98048828554 p^{9} T^{15} + 8284116752 p^{12} T^{16} - 569874862 p^{15} T^{17} + 43410603 p^{18} T^{18} - 2482934 p^{21} T^{19} + 183722 p^{24} T^{20} - 7098 p^{27} T^{21} + 557 p^{30} T^{22} - 2 p^{34} T^{23} + p^{36} T^{24} \) | |
| 7 | \( 1 - 44 T + 2272 T^{2} - 65500 T^{3} + 286646 p T^{4} - 43893396 T^{5} + 149543584 p T^{6} - 18698971716 T^{7} + 393898935151 T^{8} - 6092324543608 T^{9} + 124670988749504 T^{10} - 262645183843432 p T^{11} + 40044276129164396 T^{12} - 262645183843432 p^{4} T^{13} + 124670988749504 p^{6} T^{14} - 6092324543608 p^{9} T^{15} + 393898935151 p^{12} T^{16} - 18698971716 p^{15} T^{17} + 149543584 p^{19} T^{18} - 43893396 p^{21} T^{19} + 286646 p^{25} T^{20} - 65500 p^{27} T^{21} + 2272 p^{30} T^{22} - 44 p^{33} T^{23} + p^{36} T^{24} \) | |
| 11 | \( 1 + 46 T + 7211 T^{2} + 229414 T^{3} + 23069366 T^{4} + 479228230 T^{5} + 44120621525 T^{6} + 350584215358 T^{7} + 55076050683888 T^{8} - 624442000075682 T^{9} + 49298894587690015 T^{10} - 2117861433164375546 T^{11} + 47965568051473757908 T^{12} - 2117861433164375546 p^{3} T^{13} + 49298894587690015 p^{6} T^{14} - 624442000075682 p^{9} T^{15} + 55076050683888 p^{12} T^{16} + 350584215358 p^{15} T^{17} + 44120621525 p^{18} T^{18} + 479228230 p^{21} T^{19} + 23069366 p^{24} T^{20} + 229414 p^{27} T^{21} + 7211 p^{30} T^{22} + 46 p^{33} T^{23} + p^{36} T^{24} \) | |
| 13 | \( 1 - 34 T + 11117 T^{2} - 421906 T^{3} + 64557498 T^{4} - 2444964878 T^{5} + 260092948379 T^{6} - 9286552251014 T^{7} + 822232175370448 T^{8} - 26971836331388034 T^{9} + 2192385636044052521 T^{10} - 66299529828989512002 T^{11} + \)\(51\!\cdots\!80\)\( T^{12} - 66299529828989512002 p^{3} T^{13} + 2192385636044052521 p^{6} T^{14} - 26971836331388034 p^{9} T^{15} + 822232175370448 p^{12} T^{16} - 9286552251014 p^{15} T^{17} + 260092948379 p^{18} T^{18} - 2444964878 p^{21} T^{19} + 64557498 p^{24} T^{20} - 421906 p^{27} T^{21} + 11117 p^{30} T^{22} - 34 p^{33} T^{23} + p^{36} T^{24} \) | |
| 17 | \( 1 - 36 T + 29216 T^{2} - 715180 T^{3} + 444306106 T^{4} - 7318441692 T^{5} + 4631077579936 T^{6} - 51131776398388 T^{7} + 37185185431570511 T^{8} - 296816088323149480 T^{9} + \)\(24\!\cdots\!44\)\( T^{10} - \)\(15\!\cdots\!36\)\( T^{11} + \)\(13\!\cdots\!84\)\( T^{12} - \)\(15\!\cdots\!36\)\( p^{3} T^{13} + \)\(24\!\cdots\!44\)\( p^{6} T^{14} - 296816088323149480 p^{9} T^{15} + 37185185431570511 p^{12} T^{16} - 51131776398388 p^{15} T^{17} + 4631077579936 p^{18} T^{18} - 7318441692 p^{21} T^{19} + 444306106 p^{24} T^{20} - 715180 p^{27} T^{21} + 29216 p^{30} T^{22} - 36 p^{33} T^{23} + p^{36} T^{24} \) | |
| 19 | \( 1 + 148 T + 36704 T^{2} + 4766580 T^{3} + 692535866 T^{4} + 78760221036 T^{5} + 474086411488 p T^{6} + 883796141025388 T^{7} + 88357295286635039 T^{8} + 7753230805252515528 T^{9} + \)\(70\!\cdots\!72\)\( T^{10} + \)\(58\!\cdots\!20\)\( T^{11} + \)\(50\!\cdots\!16\)\( T^{12} + \)\(58\!\cdots\!20\)\( p^{3} T^{13} + \)\(70\!\cdots\!72\)\( p^{6} T^{14} + 7753230805252515528 p^{9} T^{15} + 88357295286635039 p^{12} T^{16} + 883796141025388 p^{15} T^{17} + 474086411488 p^{19} T^{18} + 78760221036 p^{21} T^{19} + 692535866 p^{24} T^{20} + 4766580 p^{27} T^{21} + 36704 p^{30} T^{22} + 148 p^{33} T^{23} + p^{36} T^{24} \) | |
| 23 | \( 1 - 328 T + 125060 T^{2} - 26335080 T^{3} + 5973685298 T^{4} - 942058117560 T^{5} + 161058634112436 T^{6} - 20558753674066136 T^{7} + 128404418827910025 p T^{8} - \)\(32\!\cdots\!52\)\( T^{9} + \)\(42\!\cdots\!96\)\( T^{10} - \)\(42\!\cdots\!08\)\( T^{11} + \)\(53\!\cdots\!20\)\( T^{12} - \)\(42\!\cdots\!08\)\( p^{3} T^{13} + \)\(42\!\cdots\!96\)\( p^{6} T^{14} - \)\(32\!\cdots\!52\)\( p^{9} T^{15} + 128404418827910025 p^{13} T^{16} - 20558753674066136 p^{15} T^{17} + 161058634112436 p^{18} T^{18} - 942058117560 p^{21} T^{19} + 5973685298 p^{24} T^{20} - 26335080 p^{27} T^{21} + 125060 p^{30} T^{22} - 328 p^{33} T^{23} + p^{36} T^{24} \) | |
| 31 | \( 1 - 374 T + 277651 T^{2} - 78493942 T^{3} + 34589547662 T^{4} - 8080408657262 T^{5} + 2704488610203973 T^{6} - 545084263829379470 T^{7} + \)\(15\!\cdots\!12\)\( T^{8} - \)\(26\!\cdots\!26\)\( T^{9} + \)\(64\!\cdots\!39\)\( T^{10} - \)\(10\!\cdots\!18\)\( T^{11} + \)\(21\!\cdots\!76\)\( T^{12} - \)\(10\!\cdots\!18\)\( p^{3} T^{13} + \)\(64\!\cdots\!39\)\( p^{6} T^{14} - \)\(26\!\cdots\!26\)\( p^{9} T^{15} + \)\(15\!\cdots\!12\)\( p^{12} T^{16} - 545084263829379470 p^{15} T^{17} + 2704488610203973 p^{18} T^{18} - 8080408657262 p^{21} T^{19} + 34589547662 p^{24} T^{20} - 78493942 p^{27} T^{21} + 277651 p^{30} T^{22} - 374 p^{33} T^{23} + p^{36} T^{24} \) | |
| 37 | \( 1 - 340 T + 206240 T^{2} - 45729516 T^{3} + 22818312666 T^{4} - 4911275427244 T^{5} + 2205086239808480 T^{6} - 439245666986366644 T^{7} + \)\(16\!\cdots\!91\)\( T^{8} - \)\(29\!\cdots\!52\)\( T^{9} + \)\(10\!\cdots\!96\)\( T^{10} - \)\(17\!\cdots\!64\)\( T^{11} + \)\(57\!\cdots\!64\)\( T^{12} - \)\(17\!\cdots\!64\)\( p^{3} T^{13} + \)\(10\!\cdots\!96\)\( p^{6} T^{14} - \)\(29\!\cdots\!52\)\( p^{9} T^{15} + \)\(16\!\cdots\!91\)\( p^{12} T^{16} - 439245666986366644 p^{15} T^{17} + 2205086239808480 p^{18} T^{18} - 4911275427244 p^{21} T^{19} + 22818312666 p^{24} T^{20} - 45729516 p^{27} T^{21} + 206240 p^{30} T^{22} - 340 p^{33} T^{23} + p^{36} T^{24} \) | |
| 41 | \( 1 - 32 T + 363236 T^{2} - 7534976 T^{3} + 69809153522 T^{4} - 11444927744 T^{5} + 9545321015032468 T^{6} + 154531727003197664 T^{7} + \)\(10\!\cdots\!35\)\( T^{8} + \)\(23\!\cdots\!56\)\( T^{9} + \)\(93\!\cdots\!44\)\( T^{10} + \)\(21\!\cdots\!56\)\( T^{11} + \)\(70\!\cdots\!56\)\( T^{12} + \)\(21\!\cdots\!56\)\( p^{3} T^{13} + \)\(93\!\cdots\!44\)\( p^{6} T^{14} + \)\(23\!\cdots\!56\)\( p^{9} T^{15} + \)\(10\!\cdots\!35\)\( p^{12} T^{16} + 154531727003197664 p^{15} T^{17} + 9545321015032468 p^{18} T^{18} - 11444927744 p^{21} T^{19} + 69809153522 p^{24} T^{20} - 7534976 p^{27} T^{21} + 363236 p^{30} T^{22} - 32 p^{33} T^{23} + p^{36} T^{24} \) | |
| 43 | \( 1 + 462 T + 7889 p T^{2} + 116170598 T^{3} + 60905074198 T^{4} + 17901636145158 T^{5} + 8332518075870309 T^{6} + 2370403073966505950 T^{7} + \)\(98\!\cdots\!52\)\( T^{8} + \)\(25\!\cdots\!66\)\( T^{9} + \)\(97\!\cdots\!47\)\( T^{10} + \)\(23\!\cdots\!30\)\( T^{11} + \)\(82\!\cdots\!56\)\( T^{12} + \)\(23\!\cdots\!30\)\( p^{3} T^{13} + \)\(97\!\cdots\!47\)\( p^{6} T^{14} + \)\(25\!\cdots\!66\)\( p^{9} T^{15} + \)\(98\!\cdots\!52\)\( p^{12} T^{16} + 2370403073966505950 p^{15} T^{17} + 8332518075870309 p^{18} T^{18} + 17901636145158 p^{21} T^{19} + 60905074198 p^{24} T^{20} + 116170598 p^{27} T^{21} + 7889 p^{31} T^{22} + 462 p^{33} T^{23} + p^{36} T^{24} \) | |
| 47 | \( 1 - 434 T + 547731 T^{2} - 251692834 T^{3} + 173061571134 T^{4} - 69011658725754 T^{5} + 37270175000174037 T^{6} - 12858980778278714938 T^{7} + \)\(58\!\cdots\!40\)\( T^{8} - \)\(18\!\cdots\!06\)\( T^{9} + \)\(74\!\cdots\!67\)\( T^{10} - \)\(21\!\cdots\!18\)\( T^{11} + \)\(82\!\cdots\!52\)\( T^{12} - \)\(21\!\cdots\!18\)\( p^{3} T^{13} + \)\(74\!\cdots\!67\)\( p^{6} T^{14} - \)\(18\!\cdots\!06\)\( p^{9} T^{15} + \)\(58\!\cdots\!40\)\( p^{12} T^{16} - 12858980778278714938 p^{15} T^{17} + 37270175000174037 p^{18} T^{18} - 69011658725754 p^{21} T^{19} + 173061571134 p^{24} T^{20} - 251692834 p^{27} T^{21} + 547731 p^{30} T^{22} - 434 p^{33} T^{23} + p^{36} T^{24} \) | |
| 53 | \( 1 + 610 T + 965717 T^{2} + 408904362 T^{3} + 411687476346 T^{4} + 138268466785878 T^{5} + 117610244866911555 T^{6} + 33529562934164940582 T^{7} + \)\(26\!\cdots\!64\)\( T^{8} + \)\(65\!\cdots\!54\)\( T^{9} + \)\(48\!\cdots\!49\)\( T^{10} + \)\(11\!\cdots\!90\)\( T^{11} + \)\(77\!\cdots\!88\)\( T^{12} + \)\(11\!\cdots\!90\)\( p^{3} T^{13} + \)\(48\!\cdots\!49\)\( p^{6} T^{14} + \)\(65\!\cdots\!54\)\( p^{9} T^{15} + \)\(26\!\cdots\!64\)\( p^{12} T^{16} + 33529562934164940582 p^{15} T^{17} + 117610244866911555 p^{18} T^{18} + 138268466785878 p^{21} T^{19} + 411687476346 p^{24} T^{20} + 408904362 p^{27} T^{21} + 965717 p^{30} T^{22} + 610 p^{33} T^{23} + p^{36} T^{24} \) | |
| 59 | \( 1 + 1240 T + 1494004 T^{2} + 1268005784 T^{3} + 995127834402 T^{4} + 673701480812904 T^{5} + 437743237110523076 T^{6} + \)\(25\!\cdots\!40\)\( T^{7} + \)\(14\!\cdots\!95\)\( T^{8} + \)\(78\!\cdots\!40\)\( T^{9} + \)\(40\!\cdots\!08\)\( T^{10} + \)\(19\!\cdots\!40\)\( T^{11} + \)\(90\!\cdots\!00\)\( T^{12} + \)\(19\!\cdots\!40\)\( p^{3} T^{13} + \)\(40\!\cdots\!08\)\( p^{6} T^{14} + \)\(78\!\cdots\!40\)\( p^{9} T^{15} + \)\(14\!\cdots\!95\)\( p^{12} T^{16} + \)\(25\!\cdots\!40\)\( p^{15} T^{17} + 437743237110523076 p^{18} T^{18} + 673701480812904 p^{21} T^{19} + 995127834402 p^{24} T^{20} + 1268005784 p^{27} T^{21} + 1494004 p^{30} T^{22} + 1240 p^{33} T^{23} + p^{36} T^{24} \) | |
| 61 | \( 1 - 1228 T + 2603712 T^{2} - 2521035348 T^{3} + 3054249072874 T^{4} - 2442691811237620 T^{5} + 2174041179586884800 T^{6} - \)\(14\!\cdots\!96\)\( T^{7} + \)\(10\!\cdots\!79\)\( T^{8} - \)\(61\!\cdots\!64\)\( T^{9} + \)\(37\!\cdots\!84\)\( T^{10} - \)\(18\!\cdots\!08\)\( T^{11} + \)\(97\!\cdots\!88\)\( T^{12} - \)\(18\!\cdots\!08\)\( p^{3} T^{13} + \)\(37\!\cdots\!84\)\( p^{6} T^{14} - \)\(61\!\cdots\!64\)\( p^{9} T^{15} + \)\(10\!\cdots\!79\)\( p^{12} T^{16} - \)\(14\!\cdots\!96\)\( p^{15} T^{17} + 2174041179586884800 p^{18} T^{18} - 2442691811237620 p^{21} T^{19} + 3054249072874 p^{24} T^{20} - 2521035348 p^{27} T^{21} + 2603712 p^{30} T^{22} - 1228 p^{33} T^{23} + p^{36} T^{24} \) | |
| 67 | \( 1 + 1672 T + 3064468 T^{2} + 3370749640 T^{3} + 3811267209042 T^{4} + 3311216657646968 T^{5} + 2911652400382956964 T^{6} + \)\(21\!\cdots\!48\)\( T^{7} + \)\(15\!\cdots\!67\)\( T^{8} + \)\(10\!\cdots\!72\)\( T^{9} + \)\(66\!\cdots\!36\)\( T^{10} + \)\(38\!\cdots\!84\)\( T^{11} + \)\(22\!\cdots\!16\)\( T^{12} + \)\(38\!\cdots\!84\)\( p^{3} T^{13} + \)\(66\!\cdots\!36\)\( p^{6} T^{14} + \)\(10\!\cdots\!72\)\( p^{9} T^{15} + \)\(15\!\cdots\!67\)\( p^{12} T^{16} + \)\(21\!\cdots\!48\)\( p^{15} T^{17} + 2911652400382956964 p^{18} T^{18} + 3311216657646968 p^{21} T^{19} + 3811267209042 p^{24} T^{20} + 3370749640 p^{27} T^{21} + 3064468 p^{30} T^{22} + 1672 p^{33} T^{23} + p^{36} T^{24} \) | |
| 71 | \( 1 - 3220 T + 7244144 T^{2} - 11926428996 T^{3} + 16416329153194 T^{4} - 19229506028817420 T^{5} + 19998431884755691888 T^{6} - \)\(18\!\cdots\!28\)\( T^{7} + \)\(15\!\cdots\!91\)\( T^{8} - \)\(12\!\cdots\!20\)\( T^{9} + \)\(88\!\cdots\!76\)\( T^{10} - \)\(59\!\cdots\!56\)\( T^{11} + \)\(36\!\cdots\!12\)\( T^{12} - \)\(59\!\cdots\!56\)\( p^{3} T^{13} + \)\(88\!\cdots\!76\)\( p^{6} T^{14} - \)\(12\!\cdots\!20\)\( p^{9} T^{15} + \)\(15\!\cdots\!91\)\( p^{12} T^{16} - \)\(18\!\cdots\!28\)\( p^{15} T^{17} + 19998431884755691888 p^{18} T^{18} - 19229506028817420 p^{21} T^{19} + 16416329153194 p^{24} T^{20} - 11926428996 p^{27} T^{21} + 7244144 p^{30} T^{22} - 3220 p^{33} T^{23} + p^{36} T^{24} \) | |
| 73 | \( 1 - 564 T + 2076624 T^{2} - 873085660 T^{3} + 1991640801610 T^{4} - 732713611150188 T^{5} + 1266762864267955024 T^{6} - \)\(49\!\cdots\!20\)\( T^{7} + \)\(62\!\cdots\!91\)\( T^{8} - \)\(28\!\cdots\!00\)\( T^{9} + \)\(26\!\cdots\!36\)\( T^{10} - \)\(14\!\cdots\!16\)\( T^{11} + \)\(10\!\cdots\!28\)\( T^{12} - \)\(14\!\cdots\!16\)\( p^{3} T^{13} + \)\(26\!\cdots\!36\)\( p^{6} T^{14} - \)\(28\!\cdots\!00\)\( p^{9} T^{15} + \)\(62\!\cdots\!91\)\( p^{12} T^{16} - \)\(49\!\cdots\!20\)\( p^{15} T^{17} + 1266762864267955024 p^{18} T^{18} - 732713611150188 p^{21} T^{19} + 1991640801610 p^{24} T^{20} - 873085660 p^{27} T^{21} + 2076624 p^{30} T^{22} - 564 p^{33} T^{23} + p^{36} T^{24} \) | |
| 79 | \( 1 - 1862 T + 6158443 T^{2} - 8684213622 T^{3} + 16390953535470 T^{4} - 18692108490922366 T^{5} + 25843107537407809821 T^{6} - \)\(24\!\cdots\!74\)\( T^{7} + \)\(27\!\cdots\!68\)\( T^{8} - \)\(22\!\cdots\!34\)\( T^{9} + \)\(20\!\cdots\!23\)\( T^{10} - \)\(14\!\cdots\!86\)\( T^{11} + \)\(11\!\cdots\!88\)\( T^{12} - \)\(14\!\cdots\!86\)\( p^{3} T^{13} + \)\(20\!\cdots\!23\)\( p^{6} T^{14} - \)\(22\!\cdots\!34\)\( p^{9} T^{15} + \)\(27\!\cdots\!68\)\( p^{12} T^{16} - \)\(24\!\cdots\!74\)\( p^{15} T^{17} + 25843107537407809821 p^{18} T^{18} - 18692108490922366 p^{21} T^{19} + 16390953535470 p^{24} T^{20} - 8684213622 p^{27} T^{21} + 6158443 p^{30} T^{22} - 1862 p^{33} T^{23} + p^{36} T^{24} \) | |
| 83 | \( 1 + 3736 T + 10584756 T^{2} + 21406351768 T^{3} + 36936552529474 T^{4} + 53445278020994344 T^{5} + 69113745158751616580 T^{6} + \)\(79\!\cdots\!36\)\( T^{7} + \)\(83\!\cdots\!51\)\( T^{8} + \)\(79\!\cdots\!00\)\( T^{9} + \)\(86\!\cdots\!84\)\( p T^{10} + \)\(59\!\cdots\!04\)\( T^{11} + \)\(46\!\cdots\!08\)\( T^{12} + \)\(59\!\cdots\!04\)\( p^{3} T^{13} + \)\(86\!\cdots\!84\)\( p^{7} T^{14} + \)\(79\!\cdots\!00\)\( p^{9} T^{15} + \)\(83\!\cdots\!51\)\( p^{12} T^{16} + \)\(79\!\cdots\!36\)\( p^{15} T^{17} + 69113745158751616580 p^{18} T^{18} + 53445278020994344 p^{21} T^{19} + 36936552529474 p^{24} T^{20} + 21406351768 p^{27} T^{21} + 10584756 p^{30} T^{22} + 3736 p^{33} T^{23} + p^{36} T^{24} \) | |
| 89 | \( 1 - 584 T + 4960036 T^{2} - 2906577304 T^{3} + 12290899217074 T^{4} - 7046762239854648 T^{5} + 228095085057513588 p T^{6} - \)\(11\!\cdots\!84\)\( T^{7} + \)\(25\!\cdots\!59\)\( T^{8} - \)\(13\!\cdots\!44\)\( T^{9} + \)\(24\!\cdots\!92\)\( T^{10} - \)\(11\!\cdots\!36\)\( T^{11} + \)\(18\!\cdots\!36\)\( T^{12} - \)\(11\!\cdots\!36\)\( p^{3} T^{13} + \)\(24\!\cdots\!92\)\( p^{6} T^{14} - \)\(13\!\cdots\!44\)\( p^{9} T^{15} + \)\(25\!\cdots\!59\)\( p^{12} T^{16} - \)\(11\!\cdots\!84\)\( p^{15} T^{17} + 228095085057513588 p^{19} T^{18} - 7046762239854648 p^{21} T^{19} + 12290899217074 p^{24} T^{20} - 2906577304 p^{27} T^{21} + 4960036 p^{30} T^{22} - 584 p^{33} T^{23} + p^{36} T^{24} \) | |
| 97 | \( 1 - 904 T + 5173060 T^{2} - 4844741080 T^{3} + 14440829183762 T^{4} - 13323187991321528 T^{5} + 28425088770291315380 T^{6} - \)\(24\!\cdots\!24\)\( T^{7} + \)\(43\!\cdots\!19\)\( T^{8} - \)\(34\!\cdots\!36\)\( T^{9} + \)\(52\!\cdots\!92\)\( T^{10} - \)\(39\!\cdots\!80\)\( T^{11} + \)\(52\!\cdots\!76\)\( T^{12} - \)\(39\!\cdots\!80\)\( p^{3} T^{13} + \)\(52\!\cdots\!92\)\( p^{6} T^{14} - \)\(34\!\cdots\!36\)\( p^{9} T^{15} + \)\(43\!\cdots\!19\)\( p^{12} T^{16} - \)\(24\!\cdots\!24\)\( p^{15} T^{17} + 28425088770291315380 p^{18} T^{18} - 13323187991321528 p^{21} T^{19} + 14440829183762 p^{24} T^{20} - 4844741080 p^{27} T^{21} + 5173060 p^{30} T^{22} - 904 p^{33} T^{23} + p^{36} T^{24} \) | |
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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
−2.47612931332895165033923645918, −2.16704098280529998691943726945, −2.15715212984364516338003840908, −2.14333563229842521301864379540, −2.03265652399136662840305961938, −1.89160107810222201524562307805, −1.88338743929966165094882983550, −1.76654340347678687381151072374, −1.75299680043428732461771447213, −1.68185257868308230089828628717, −1.67701041821813343924201487379, −1.43431315216554725002984207038, −1.37959430839094073395948660818, −1.22535235894049965392634271197, −1.13371991456035373180783422780, −0.831663111702098136619886812972, −0.74791970852240104684473045343, −0.70360611634289038872795397778, −0.64825932977477679299733350988, −0.62904631213729373091509526807, −0.57883646819419286459492220983, −0.33811193433676434060361259311, −0.21416710628756728758618672224, −0.19689741495151284776369080391, −0.19542659623613621642595570859, 0.19542659623613621642595570859, 0.19689741495151284776369080391, 0.21416710628756728758618672224, 0.33811193433676434060361259311, 0.57883646819419286459492220983, 0.62904631213729373091509526807, 0.64825932977477679299733350988, 0.70360611634289038872795397778, 0.74791970852240104684473045343, 0.831663111702098136619886812972, 1.13371991456035373180783422780, 1.22535235894049965392634271197, 1.37959430839094073395948660818, 1.43431315216554725002984207038, 1.67701041821813343924201487379, 1.68185257868308230089828628717, 1.75299680043428732461771447213, 1.76654340347678687381151072374, 1.88338743929966165094882983550, 1.89160107810222201524562307805, 2.03265652399136662840305961938, 2.14333563229842521301864379540, 2.15715212984364516338003840908, 2.16704098280529998691943726945, 2.47612931332895165033923645918
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.