Dirichlet series
| L(s) = 1 | − 6·2-s − 12·4-s − 12·5-s − 42·7-s + 153·8-s + 72·10-s − 42·11-s − 78·13-s + 252·14-s − 150·16-s + 18·17-s − 228·19-s + 144·20-s + 252·22-s + 114·23-s − 687·25-s + 468·26-s + 504·28-s + 660·29-s − 708·31-s − 1.54e3·32-s − 108·34-s + 504·35-s − 354·37-s + 1.36e3·38-s − 1.83e3·40-s + 1.03e3·41-s + ⋯ |
| L(s) = 1 | − 2.12·2-s − 3/2·4-s − 1.07·5-s − 2.26·7-s + 6.76·8-s + 2.27·10-s − 1.15·11-s − 1.66·13-s + 4.81·14-s − 2.34·16-s + 0.256·17-s − 2.75·19-s + 1.60·20-s + 2.44·22-s + 1.03·23-s − 5.49·25-s + 3.53·26-s + 3.40·28-s + 4.22·29-s − 4.10·31-s − 8.53·32-s − 0.544·34-s + 2.43·35-s − 1.57·37-s + 5.83·38-s − 7.25·40-s + 3.93·41-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{72}\right)^{s/2} \, \Gamma_{\C}(s)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{72}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(24\) |
| Conductor: | \(3^{72}\) |
| Sign: | $1$ |
| Analytic conductor: | \(4.00980\times 10^{19}\) |
| Root analytic conductor: | \(6.55838\) |
| Motivic weight: | \(3\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(12\) |
| Selberg data: | \((24,\ 3^{72} ,\ ( \ : [3/2]^{12} ),\ 1 )\) |
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 | \( 1 \) |
| good | 2 | \( 1 + 3 p T + 3 p^{4} T^{2} + 207 T^{3} + 525 p T^{4} + 3885 T^{5} + 8077 p T^{6} + 55377 T^{7} + 201177 T^{8} + 635877 T^{9} + 1026261 p T^{10} + 747189 p^{3} T^{11} + 2205423 p^{3} T^{12} + 747189 p^{6} T^{13} + 1026261 p^{7} T^{14} + 635877 p^{9} T^{15} + 201177 p^{12} T^{16} + 55377 p^{15} T^{17} + 8077 p^{19} T^{18} + 3885 p^{21} T^{19} + 525 p^{25} T^{20} + 207 p^{27} T^{21} + 3 p^{34} T^{22} + 3 p^{34} T^{23} + p^{36} T^{24} \) |
| 5 | \( 1 + 12 T + 831 T^{2} + 18 p^{4} T^{3} + 369699 T^{4} + 5062422 T^{5} + 113589566 T^{6} + 1473568758 T^{7} + 26132727888 T^{8} + 310624976034 T^{9} + 4660139652948 T^{10} + 49912371950148 T^{11} + 654659995378509 T^{12} + 49912371950148 p^{3} T^{13} + 4660139652948 p^{6} T^{14} + 310624976034 p^{9} T^{15} + 26132727888 p^{12} T^{16} + 1473568758 p^{15} T^{17} + 113589566 p^{18} T^{18} + 5062422 p^{21} T^{19} + 369699 p^{24} T^{20} + 18 p^{31} T^{21} + 831 p^{30} T^{22} + 12 p^{33} T^{23} + p^{36} T^{24} \) | |
| 7 | \( 1 + 6 p T + 3036 T^{2} + 101842 T^{3} + 615315 p T^{4} + 121274466 T^{5} + 3874201291 T^{6} + 94568685912 T^{7} + 2503702606005 T^{8} + 53969769936898 T^{9} + 176352850122207 p T^{10} + 23664619067852244 T^{11} + 476742898558106410 T^{12} + 23664619067852244 p^{3} T^{13} + 176352850122207 p^{7} T^{14} + 53969769936898 p^{9} T^{15} + 2503702606005 p^{12} T^{16} + 94568685912 p^{15} T^{17} + 3874201291 p^{18} T^{18} + 121274466 p^{21} T^{19} + 615315 p^{25} T^{20} + 101842 p^{27} T^{21} + 3036 p^{30} T^{22} + 6 p^{34} T^{23} + p^{36} T^{24} \) | |
| 11 | \( 1 + 42 T + 10938 T^{2} + 399708 T^{3} + 57060717 T^{4} + 1846526892 T^{5} + 190126522637 T^{6} + 5522021579286 T^{7} + 457335567708237 T^{8} + 12021670470373236 T^{9} + 848270152906356753 T^{10} + 20203589745149563296 T^{11} + \)\(12\!\cdots\!18\)\( T^{12} + 20203589745149563296 p^{3} T^{13} + 848270152906356753 p^{6} T^{14} + 12021670470373236 p^{9} T^{15} + 457335567708237 p^{12} T^{16} + 5522021579286 p^{15} T^{17} + 190126522637 p^{18} T^{18} + 1846526892 p^{21} T^{19} + 57060717 p^{24} T^{20} + 399708 p^{27} T^{21} + 10938 p^{30} T^{22} + 42 p^{33} T^{23} + p^{36} T^{24} \) | |
| 13 | \( 1 + 6 p T + 17598 T^{2} + 1171762 T^{3} + 144122232 T^{4} + 8131311312 T^{5} + 727323282172 T^{6} + 35054502969726 T^{7} + 2575226035123182 T^{8} + 108478974509515420 T^{9} + 7080074634468856686 T^{10} + \)\(27\!\cdots\!58\)\( T^{11} + \)\(16\!\cdots\!81\)\( T^{12} + \)\(27\!\cdots\!58\)\( p^{3} T^{13} + 7080074634468856686 p^{6} T^{14} + 108478974509515420 p^{9} T^{15} + 2575226035123182 p^{12} T^{16} + 35054502969726 p^{15} T^{17} + 727323282172 p^{18} T^{18} + 8131311312 p^{21} T^{19} + 144122232 p^{24} T^{20} + 1171762 p^{27} T^{21} + 17598 p^{30} T^{22} + 6 p^{34} T^{23} + p^{36} T^{24} \) | |
| 17 | \( 1 - 18 T + 1605 p T^{2} - 46188 T^{3} + 378799590 T^{4} + 3970670040 T^{5} + 3687982052216 T^{6} + 68480154333612 T^{7} + 28334851651381545 T^{8} + 642470064566247468 T^{9} + \)\(17\!\cdots\!23\)\( T^{10} + \)\(42\!\cdots\!06\)\( T^{11} + \)\(56\!\cdots\!01\)\( p T^{12} + \)\(42\!\cdots\!06\)\( p^{3} T^{13} + \)\(17\!\cdots\!23\)\( p^{6} T^{14} + 642470064566247468 p^{9} T^{15} + 28334851651381545 p^{12} T^{16} + 68480154333612 p^{15} T^{17} + 3687982052216 p^{18} T^{18} + 3970670040 p^{21} T^{19} + 378799590 p^{24} T^{20} - 46188 p^{27} T^{21} + 1605 p^{31} T^{22} - 18 p^{33} T^{23} + p^{36} T^{24} \) | |
| 19 | \( 1 + 12 p T + 65598 T^{2} + 10557550 T^{3} + 1895939457 T^{4} + 244909953084 T^{5} + 34002829867213 T^{6} + 3727551684296016 T^{7} + 433911156710034501 T^{8} + 41620379928293229856 T^{9} + \)\(42\!\cdots\!61\)\( T^{10} + \)\(18\!\cdots\!26\)\( p T^{11} + \)\(32\!\cdots\!50\)\( T^{12} + \)\(18\!\cdots\!26\)\( p^{4} T^{13} + \)\(42\!\cdots\!61\)\( p^{6} T^{14} + 41620379928293229856 p^{9} T^{15} + 433911156710034501 p^{12} T^{16} + 3727551684296016 p^{15} T^{17} + 34002829867213 p^{18} T^{18} + 244909953084 p^{21} T^{19} + 1895939457 p^{24} T^{20} + 10557550 p^{27} T^{21} + 65598 p^{30} T^{22} + 12 p^{34} T^{23} + p^{36} T^{24} \) | |
| 23 | \( 1 - 114 T + 86844 T^{2} - 7473096 T^{3} + 3589570059 T^{4} - 9934897818 p T^{5} + 94603846309235 T^{6} - 4216862110588992 T^{7} + 1815581284007638833 T^{8} - 53177452533524291454 T^{9} + \)\(27\!\cdots\!97\)\( T^{10} - \)\(55\!\cdots\!06\)\( T^{11} + \)\(36\!\cdots\!98\)\( T^{12} - \)\(55\!\cdots\!06\)\( p^{3} T^{13} + \)\(27\!\cdots\!97\)\( p^{6} T^{14} - 53177452533524291454 p^{9} T^{15} + 1815581284007638833 p^{12} T^{16} - 4216862110588992 p^{15} T^{17} + 94603846309235 p^{18} T^{18} - 9934897818 p^{22} T^{19} + 3589570059 p^{24} T^{20} - 7473096 p^{27} T^{21} + 86844 p^{30} T^{22} - 114 p^{33} T^{23} + p^{36} T^{24} \) | |
| 29 | \( 1 - 660 T + 333813 T^{2} - 118267254 T^{3} + 36609873711 T^{4} - 9440888603268 T^{5} + 2231799317094842 T^{6} - 466678842948720126 T^{7} + 92048488913551542102 T^{8} - \)\(16\!\cdots\!54\)\( T^{9} + \)\(28\!\cdots\!88\)\( T^{10} - \)\(47\!\cdots\!52\)\( T^{11} + \)\(75\!\cdots\!57\)\( T^{12} - \)\(47\!\cdots\!52\)\( p^{3} T^{13} + \)\(28\!\cdots\!88\)\( p^{6} T^{14} - \)\(16\!\cdots\!54\)\( p^{9} T^{15} + 92048488913551542102 p^{12} T^{16} - 466678842948720126 p^{15} T^{17} + 2231799317094842 p^{18} T^{18} - 9440888603268 p^{21} T^{19} + 36609873711 p^{24} T^{20} - 118267254 p^{27} T^{21} + 333813 p^{30} T^{22} - 660 p^{33} T^{23} + p^{36} T^{24} \) | |
| 31 | \( 1 + 708 T + 396237 T^{2} + 152780596 T^{3} + 52458006351 T^{4} + 15058806235008 T^{5} + 4064767979251492 T^{6} + 980515278323793720 T^{7} + \)\(22\!\cdots\!57\)\( T^{8} + \)\(47\!\cdots\!52\)\( T^{9} + \)\(96\!\cdots\!11\)\( T^{10} + \)\(17\!\cdots\!68\)\( T^{11} + \)\(32\!\cdots\!70\)\( T^{12} + \)\(17\!\cdots\!68\)\( p^{3} T^{13} + \)\(96\!\cdots\!11\)\( p^{6} T^{14} + \)\(47\!\cdots\!52\)\( p^{9} T^{15} + \)\(22\!\cdots\!57\)\( p^{12} T^{16} + 980515278323793720 p^{15} T^{17} + 4064767979251492 p^{18} T^{18} + 15058806235008 p^{21} T^{19} + 52458006351 p^{24} T^{20} + 152780596 p^{27} T^{21} + 396237 p^{30} T^{22} + 708 p^{33} T^{23} + p^{36} T^{24} \) | |
| 37 | \( 1 + 354 T + 241377 T^{2} + 74455912 T^{3} + 34606095729 T^{4} + 9686749492272 T^{5} + 3561065782685848 T^{6} + 924024438254694024 T^{7} + \)\(29\!\cdots\!70\)\( T^{8} + \)\(69\!\cdots\!20\)\( T^{9} + \)\(19\!\cdots\!64\)\( T^{10} + \)\(42\!\cdots\!84\)\( T^{11} + \)\(10\!\cdots\!81\)\( T^{12} + \)\(42\!\cdots\!84\)\( p^{3} T^{13} + \)\(19\!\cdots\!64\)\( p^{6} T^{14} + \)\(69\!\cdots\!20\)\( p^{9} T^{15} + \)\(29\!\cdots\!70\)\( p^{12} T^{16} + 924024438254694024 p^{15} T^{17} + 3561065782685848 p^{18} T^{18} + 9686749492272 p^{21} T^{19} + 34606095729 p^{24} T^{20} + 74455912 p^{27} T^{21} + 241377 p^{30} T^{22} + 354 p^{33} T^{23} + p^{36} T^{24} \) | |
| 41 | \( 1 - 1032 T + 921684 T^{2} - 562850946 T^{3} + 310655218110 T^{4} - 142597486514472 T^{5} + 60722193909672899 T^{6} - 22922842456199835774 T^{7} + \)\(81\!\cdots\!35\)\( T^{8} - \)\(26\!\cdots\!24\)\( T^{9} + \)\(81\!\cdots\!62\)\( T^{10} - \)\(23\!\cdots\!34\)\( T^{11} + \)\(63\!\cdots\!91\)\( T^{12} - \)\(23\!\cdots\!34\)\( p^{3} T^{13} + \)\(81\!\cdots\!62\)\( p^{6} T^{14} - \)\(26\!\cdots\!24\)\( p^{9} T^{15} + \)\(81\!\cdots\!35\)\( p^{12} T^{16} - 22922842456199835774 p^{15} T^{17} + 60722193909672899 p^{18} T^{18} - 142597486514472 p^{21} T^{19} + 310655218110 p^{24} T^{20} - 562850946 p^{27} T^{21} + 921684 p^{30} T^{22} - 1032 p^{33} T^{23} + p^{36} T^{24} \) | |
| 43 | \( 1 + 744 T + 553692 T^{2} + 225937744 T^{3} + 104527998066 T^{4} + 33437243209776 T^{5} + 14025330871137100 T^{6} + 4278437402978673432 T^{7} + \)\(16\!\cdots\!27\)\( T^{8} + \)\(47\!\cdots\!80\)\( T^{9} + \)\(16\!\cdots\!68\)\( T^{10} + \)\(42\!\cdots\!16\)\( T^{11} + \)\(14\!\cdots\!92\)\( T^{12} + \)\(42\!\cdots\!16\)\( p^{3} T^{13} + \)\(16\!\cdots\!68\)\( p^{6} T^{14} + \)\(47\!\cdots\!80\)\( p^{9} T^{15} + \)\(16\!\cdots\!27\)\( p^{12} T^{16} + 4278437402978673432 p^{15} T^{17} + 14025330871137100 p^{18} T^{18} + 33437243209776 p^{21} T^{19} + 104527998066 p^{24} T^{20} + 225937744 p^{27} T^{21} + 553692 p^{30} T^{22} + 744 p^{33} T^{23} + p^{36} T^{24} \) | |
| 47 | \( 1 + 942 T + 1058889 T^{2} + 690880410 T^{3} + 471117013206 T^{4} + 241228679983080 T^{5} + 125419907597914160 T^{6} + 53682080815284171714 T^{7} + \)\(23\!\cdots\!13\)\( T^{8} + \)\(86\!\cdots\!44\)\( T^{9} + \)\(32\!\cdots\!71\)\( T^{10} + \)\(10\!\cdots\!46\)\( T^{11} + \)\(37\!\cdots\!60\)\( T^{12} + \)\(10\!\cdots\!46\)\( p^{3} T^{13} + \)\(32\!\cdots\!71\)\( p^{6} T^{14} + \)\(86\!\cdots\!44\)\( p^{9} T^{15} + \)\(23\!\cdots\!13\)\( p^{12} T^{16} + 53682080815284171714 p^{15} T^{17} + 125419907597914160 p^{18} T^{18} + 241228679983080 p^{21} T^{19} + 471117013206 p^{24} T^{20} + 690880410 p^{27} T^{21} + 1058889 p^{30} T^{22} + 942 p^{33} T^{23} + p^{36} T^{24} \) | |
| 53 | \( 1 - 828 T + 1410657 T^{2} - 990164700 T^{3} + 926325130575 T^{4} - 563788608990552 T^{5} + 379915391732165060 T^{6} - \)\(20\!\cdots\!52\)\( T^{7} + \)\(10\!\cdots\!85\)\( T^{8} - \)\(52\!\cdots\!16\)\( T^{9} + \)\(23\!\cdots\!59\)\( T^{10} - \)\(10\!\cdots\!92\)\( T^{11} + \)\(40\!\cdots\!58\)\( T^{12} - \)\(10\!\cdots\!92\)\( p^{3} T^{13} + \)\(23\!\cdots\!59\)\( p^{6} T^{14} - \)\(52\!\cdots\!16\)\( p^{9} T^{15} + \)\(10\!\cdots\!85\)\( p^{12} T^{16} - \)\(20\!\cdots\!52\)\( p^{15} T^{17} + 379915391732165060 p^{18} T^{18} - 563788608990552 p^{21} T^{19} + 926325130575 p^{24} T^{20} - 990164700 p^{27} T^{21} + 1410657 p^{30} T^{22} - 828 p^{33} T^{23} + p^{36} T^{24} \) | |
| 59 | \( 1 - 24 T + 1154712 T^{2} + 108879264 T^{3} + 654469055961 T^{4} + 142630153314726 T^{5} + 256843348960078295 T^{6} + 79288323473587318434 T^{7} + \)\(82\!\cdots\!25\)\( T^{8} + \)\(27\!\cdots\!06\)\( T^{9} + \)\(22\!\cdots\!05\)\( T^{10} + \)\(69\!\cdots\!94\)\( T^{11} + \)\(51\!\cdots\!18\)\( T^{12} + \)\(69\!\cdots\!94\)\( p^{3} T^{13} + \)\(22\!\cdots\!05\)\( p^{6} T^{14} + \)\(27\!\cdots\!06\)\( p^{9} T^{15} + \)\(82\!\cdots\!25\)\( p^{12} T^{16} + 79288323473587318434 p^{15} T^{17} + 256843348960078295 p^{18} T^{18} + 142630153314726 p^{21} T^{19} + 654469055961 p^{24} T^{20} + 108879264 p^{27} T^{21} + 1154712 p^{30} T^{22} - 24 p^{33} T^{23} + p^{36} T^{24} \) | |
| 61 | \( 1 + 1698 T + 2661744 T^{2} + 2733019648 T^{3} + 2588156484195 T^{4} + 1954552764603318 T^{5} + 1384169662080875359 T^{6} + \)\(83\!\cdots\!16\)\( T^{7} + \)\(48\!\cdots\!89\)\( T^{8} + \)\(24\!\cdots\!54\)\( T^{9} + \)\(12\!\cdots\!97\)\( T^{10} + \)\(59\!\cdots\!10\)\( T^{11} + \)\(28\!\cdots\!86\)\( T^{12} + \)\(59\!\cdots\!10\)\( p^{3} T^{13} + \)\(12\!\cdots\!97\)\( p^{6} T^{14} + \)\(24\!\cdots\!54\)\( p^{9} T^{15} + \)\(48\!\cdots\!89\)\( p^{12} T^{16} + \)\(83\!\cdots\!16\)\( p^{15} T^{17} + 1384169662080875359 p^{18} T^{18} + 1954552764603318 p^{21} T^{19} + 2588156484195 p^{24} T^{20} + 2733019648 p^{27} T^{21} + 2661744 p^{30} T^{22} + 1698 p^{33} T^{23} + p^{36} T^{24} \) | |
| 67 | \( 1 + 1266 T + 2242281 T^{2} + 2079332890 T^{3} + 2203397904414 T^{4} + 1667100797153556 T^{5} + 1356716020460611204 T^{6} + \)\(89\!\cdots\!98\)\( T^{7} + \)\(61\!\cdots\!05\)\( T^{8} + \)\(36\!\cdots\!48\)\( T^{9} + \)\(22\!\cdots\!27\)\( T^{10} + \)\(12\!\cdots\!78\)\( T^{11} + \)\(72\!\cdots\!60\)\( T^{12} + \)\(12\!\cdots\!78\)\( p^{3} T^{13} + \)\(22\!\cdots\!27\)\( p^{6} T^{14} + \)\(36\!\cdots\!48\)\( p^{9} T^{15} + \)\(61\!\cdots\!05\)\( p^{12} T^{16} + \)\(89\!\cdots\!98\)\( p^{15} T^{17} + 1356716020460611204 p^{18} T^{18} + 1667100797153556 p^{21} T^{19} + 2203397904414 p^{24} T^{20} + 2079332890 p^{27} T^{21} + 2242281 p^{30} T^{22} + 1266 p^{33} T^{23} + p^{36} T^{24} \) | |
| 71 | \( 1 + 3888 T + 9472848 T^{2} + 16698964968 T^{3} + 23909532632490 T^{4} + 28876815994589208 T^{5} + 30580771234603876304 T^{6} + \)\(28\!\cdots\!92\)\( T^{7} + \)\(24\!\cdots\!67\)\( T^{8} + \)\(19\!\cdots\!84\)\( T^{9} + \)\(14\!\cdots\!32\)\( T^{10} + \)\(94\!\cdots\!00\)\( T^{11} + \)\(58\!\cdots\!64\)\( T^{12} + \)\(94\!\cdots\!00\)\( p^{3} T^{13} + \)\(14\!\cdots\!32\)\( p^{6} T^{14} + \)\(19\!\cdots\!84\)\( p^{9} T^{15} + \)\(24\!\cdots\!67\)\( p^{12} T^{16} + \)\(28\!\cdots\!92\)\( p^{15} T^{17} + 30580771234603876304 p^{18} T^{18} + 28876815994589208 p^{21} T^{19} + 23909532632490 p^{24} T^{20} + 16698964968 p^{27} T^{21} + 9472848 p^{30} T^{22} + 3888 p^{33} T^{23} + p^{36} T^{24} \) | |
| 73 | \( 1 + 1164 T + 2863680 T^{2} + 2986228324 T^{3} + 4116691297176 T^{4} + 3722553087241332 T^{5} + 3877761578271121090 T^{6} + \)\(30\!\cdots\!96\)\( T^{7} + \)\(26\!\cdots\!12\)\( T^{8} + \)\(19\!\cdots\!44\)\( T^{9} + \)\(14\!\cdots\!64\)\( T^{10} + \)\(92\!\cdots\!64\)\( T^{11} + \)\(62\!\cdots\!07\)\( T^{12} + \)\(92\!\cdots\!64\)\( p^{3} T^{13} + \)\(14\!\cdots\!64\)\( p^{6} T^{14} + \)\(19\!\cdots\!44\)\( p^{9} T^{15} + \)\(26\!\cdots\!12\)\( p^{12} T^{16} + \)\(30\!\cdots\!96\)\( p^{15} T^{17} + 3877761578271121090 p^{18} T^{18} + 3722553087241332 p^{21} T^{19} + 4116691297176 p^{24} T^{20} + 2986228324 p^{27} T^{21} + 2863680 p^{30} T^{22} + 1164 p^{33} T^{23} + p^{36} T^{24} \) | |
| 79 | \( 1 + 2382 T + 6094200 T^{2} + 9584500318 T^{3} + 14733487983153 T^{4} + 17532778836000066 T^{5} + 20236880655873465859 T^{6} + \)\(19\!\cdots\!96\)\( T^{7} + \)\(18\!\cdots\!01\)\( T^{8} + \)\(15\!\cdots\!10\)\( T^{9} + \)\(12\!\cdots\!69\)\( T^{10} + \)\(91\!\cdots\!60\)\( T^{11} + \)\(68\!\cdots\!18\)\( T^{12} + \)\(91\!\cdots\!60\)\( p^{3} T^{13} + \)\(12\!\cdots\!69\)\( p^{6} T^{14} + \)\(15\!\cdots\!10\)\( p^{9} T^{15} + \)\(18\!\cdots\!01\)\( p^{12} T^{16} + \)\(19\!\cdots\!96\)\( p^{15} T^{17} + 20236880655873465859 p^{18} T^{18} + 17532778836000066 p^{21} T^{19} + 14733487983153 p^{24} T^{20} + 9584500318 p^{27} T^{21} + 6094200 p^{30} T^{22} + 2382 p^{33} T^{23} + p^{36} T^{24} \) | |
| 83 | \( 1 - 4008 T + 10837092 T^{2} - 21539693052 T^{3} + 35491475230593 T^{4} - 49939724231864238 T^{5} + 62383132568798653751 T^{6} - \)\(70\!\cdots\!46\)\( T^{7} + \)\(72\!\cdots\!73\)\( T^{8} - \)\(68\!\cdots\!66\)\( T^{9} + \)\(61\!\cdots\!09\)\( T^{10} - \)\(50\!\cdots\!58\)\( T^{11} + \)\(39\!\cdots\!94\)\( T^{12} - \)\(50\!\cdots\!58\)\( p^{3} T^{13} + \)\(61\!\cdots\!09\)\( p^{6} T^{14} - \)\(68\!\cdots\!66\)\( p^{9} T^{15} + \)\(72\!\cdots\!73\)\( p^{12} T^{16} - \)\(70\!\cdots\!46\)\( p^{15} T^{17} + 62383132568798653751 p^{18} T^{18} - 49939724231864238 p^{21} T^{19} + 35491475230593 p^{24} T^{20} - 21539693052 p^{27} T^{21} + 10837092 p^{30} T^{22} - 4008 p^{33} T^{23} + p^{36} T^{24} \) | |
| 89 | \( 1 - 3582 T + 9208023 T^{2} - 16371231642 T^{3} + 24664465339941 T^{4} - 30799525034820456 T^{5} + 35164856630730158042 T^{6} - \)\(35\!\cdots\!66\)\( T^{7} + \)\(35\!\cdots\!54\)\( T^{8} - \)\(32\!\cdots\!28\)\( T^{9} + \)\(28\!\cdots\!78\)\( T^{10} - \)\(24\!\cdots\!84\)\( T^{11} + \)\(20\!\cdots\!31\)\( T^{12} - \)\(24\!\cdots\!84\)\( p^{3} T^{13} + \)\(28\!\cdots\!78\)\( p^{6} T^{14} - \)\(32\!\cdots\!28\)\( p^{9} T^{15} + \)\(35\!\cdots\!54\)\( p^{12} T^{16} - \)\(35\!\cdots\!66\)\( p^{15} T^{17} + 35164856630730158042 p^{18} T^{18} - 30799525034820456 p^{21} T^{19} + 24664465339941 p^{24} T^{20} - 16371231642 p^{27} T^{21} + 9208023 p^{30} T^{22} - 3582 p^{33} T^{23} + p^{36} T^{24} \) | |
| 97 | \( 1 + 2958 T + 9267033 T^{2} + 17310905872 T^{3} + 32258081601246 T^{4} + 45739676328280764 T^{5} + 65493331632543782548 T^{6} + \)\(78\!\cdots\!64\)\( T^{7} + \)\(97\!\cdots\!97\)\( T^{8} + \)\(10\!\cdots\!40\)\( T^{9} + \)\(11\!\cdots\!75\)\( T^{10} + \)\(11\!\cdots\!30\)\( T^{11} + \)\(11\!\cdots\!93\)\( T^{12} + \)\(11\!\cdots\!30\)\( p^{3} T^{13} + \)\(11\!\cdots\!75\)\( p^{6} T^{14} + \)\(10\!\cdots\!40\)\( p^{9} T^{15} + \)\(97\!\cdots\!97\)\( p^{12} T^{16} + \)\(78\!\cdots\!64\)\( p^{15} T^{17} + 65493331632543782548 p^{18} T^{18} + 45739676328280764 p^{21} T^{19} + 32258081601246 p^{24} T^{20} + 17310905872 p^{27} T^{21} + 9267033 p^{30} T^{22} + 2958 p^{33} T^{23} + p^{36} T^{24} \) | |
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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
−3.42929911007811144862292066857, −3.38885142124257634644467567073, −3.37592572394990817215756295684, −3.21590534241337533782720449191, −3.08705953444681537812423322391, −2.92023081437122245038354063608, −2.91174997146627189071504703332, −2.86118011775947796808609270899, −2.67016683419126260393896804244, −2.54987245452327908161607043991, −2.52866965430446396264905710432, −2.38450245561170926897565625498, −2.28313245796351115196952545098, −2.25946163958332616887493508798, −2.02881911936424057624363202624, −1.89867807127348884656450433032, −1.54752154362656577012885012450, −1.53218522790649262392809046962, −1.52161934914990450043539727985, −1.52141209927975624102735209899, −1.29995900300356915846892043289, −1.16582936168310692570938064062, −1.11033080270694175526676275209, −1.08146117700454714254201836128, −0.867936414641235139284565861962, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0.867936414641235139284565861962, 1.08146117700454714254201836128, 1.11033080270694175526676275209, 1.16582936168310692570938064062, 1.29995900300356915846892043289, 1.52141209927975624102735209899, 1.52161934914990450043539727985, 1.53218522790649262392809046962, 1.54752154362656577012885012450, 1.89867807127348884656450433032, 2.02881911936424057624363202624, 2.25946163958332616887493508798, 2.28313245796351115196952545098, 2.38450245561170926897565625498, 2.52866965430446396264905710432, 2.54987245452327908161607043991, 2.67016683419126260393896804244, 2.86118011775947796808609270899, 2.91174997146627189071504703332, 2.92023081437122245038354063608, 3.08705953444681537812423322391, 3.21590534241337533782720449191, 3.37592572394990817215756295684, 3.38885142124257634644467567073, 3.42929911007811144862292066857
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.