Dirichlet series
| L(s) = 1 | + 4·2-s + 8·4-s + 20·5-s − 4·7-s + 12·8-s + 80·10-s − 40·11-s − 16·14-s + 94·16-s − 52·17-s − 12·19-s + 160·20-s − 160·22-s + 276·23-s − 32·25-s − 32·28-s − 632·29-s + 188·31-s + 236·32-s − 208·34-s − 80·35-s + 940·37-s − 48·38-s + 240·40-s − 176·41-s − 1.36e3·43-s − 320·44-s + ⋯ |
| L(s) = 1 | + 1.41·2-s + 4-s + 1.78·5-s − 0.215·7-s + 0.530·8-s + 2.52·10-s − 1.09·11-s − 0.305·14-s + 1.46·16-s − 0.741·17-s − 0.144·19-s + 1.78·20-s − 1.55·22-s + 2.50·23-s − 0.255·25-s − 0.215·28-s − 4.04·29-s + 1.08·31-s + 1.30·32-s − 1.04·34-s − 0.386·35-s + 4.17·37-s − 0.204·38-s + 0.948·40-s − 0.670·41-s − 4.82·43-s − 1.09·44-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{24} \cdot 17^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{24} \cdot 17^{12}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(24\) |
| Conductor: | \(3^{24} \cdot 17^{12}\) |
| Sign: | $1$ |
| Analytic conductor: | \(2.92880\times 10^{11}\) |
| Root analytic conductor: | \(3.00454\) |
| Motivic weight: | \(3\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((24,\ 3^{24} \cdot 17^{12} ,\ ( \ : [3/2]^{12} ),\ 1 )\) |
Particular Values
| \(L(2)\) | \(\approx\) | \(0.06070427579\) |
| \(L(\frac12)\) | \(\approx\) | \(0.06070427579\) |
| \(L(\frac{5}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 3 | \( 1 \) |
| 17 | \( 1 + 52 T - 72 p T^{2} - 100 p^{3} T^{3} - 5793 p^{3} T^{4} + 15312 p^{4} T^{5} + 83632 p^{5} T^{6} + 15312 p^{7} T^{7} - 5793 p^{9} T^{8} - 100 p^{12} T^{9} - 72 p^{13} T^{10} + 52 p^{15} T^{11} + p^{18} T^{12} \) | |
| good | 2 | \( 1 - p^{2} T + p^{3} T^{2} - 3 p^{2} T^{3} - 31 p T^{4} + 97 p^{2} T^{5} - 123 p^{3} T^{6} + 925 p^{2} T^{7} - 6783 T^{8} - 815 p^{3} T^{9} + 1569 p^{5} T^{10} - 1649 p^{7} T^{11} + 13335 p^{6} T^{12} - 1649 p^{10} T^{13} + 1569 p^{11} T^{14} - 815 p^{12} T^{15} - 6783 p^{12} T^{16} + 925 p^{17} T^{17} - 123 p^{21} T^{18} + 97 p^{23} T^{19} - 31 p^{25} T^{20} - 3 p^{29} T^{21} + p^{33} T^{22} - p^{35} T^{23} + p^{36} T^{24} \) |
| 5 | \( 1 - 4 p T + 432 T^{2} - 5812 T^{3} + 76752 T^{4} - 134348 p T^{5} + 219872 p^{2} T^{6} - 43772 p^{3} T^{7} - 426046641 T^{8} + 1841924144 p T^{9} - 139560868176 T^{10} + 1662905986592 T^{11} - 19002322495072 T^{12} + 1662905986592 p^{3} T^{13} - 139560868176 p^{6} T^{14} + 1841924144 p^{10} T^{15} - 426046641 p^{12} T^{16} - 43772 p^{18} T^{17} + 219872 p^{20} T^{18} - 134348 p^{22} T^{19} + 76752 p^{24} T^{20} - 5812 p^{27} T^{21} + 432 p^{30} T^{22} - 4 p^{34} T^{23} + p^{36} T^{24} \) | |
| 7 | \( 1 + 4 T - 514 T^{2} - 6116 T^{3} + 115922 T^{4} + 1116940 T^{5} - 8154266 T^{6} + 68283836 p T^{7} + 5461415235 T^{8} - 238144861440 T^{9} - 823268249892 p T^{10} + 45937354375984 T^{11} + 1745889417070628 T^{12} + 45937354375984 p^{3} T^{13} - 823268249892 p^{7} T^{14} - 238144861440 p^{9} T^{15} + 5461415235 p^{12} T^{16} + 68283836 p^{16} T^{17} - 8154266 p^{18} T^{18} + 1116940 p^{21} T^{19} + 115922 p^{24} T^{20} - 6116 p^{27} T^{21} - 514 p^{30} T^{22} + 4 p^{33} T^{23} + p^{36} T^{24} \) | |
| 11 | \( 1 + 40 T + 4148 T^{2} + 162224 T^{3} + 9095112 T^{4} + 332343560 T^{5} + 18025759684 T^{6} + 645155164752 T^{7} + 32705329538127 T^{8} + 1145801951121704 T^{9} + 50413798046517864 T^{10} + 1580384424194772536 T^{11} + 69762836759754864688 T^{12} + 1580384424194772536 p^{3} T^{13} + 50413798046517864 p^{6} T^{14} + 1145801951121704 p^{9} T^{15} + 32705329538127 p^{12} T^{16} + 645155164752 p^{15} T^{17} + 18025759684 p^{18} T^{18} + 332343560 p^{21} T^{19} + 9095112 p^{24} T^{20} + 162224 p^{27} T^{21} + 4148 p^{30} T^{22} + 40 p^{33} T^{23} + p^{36} T^{24} \) | |
| 13 | \( 1 - 14448 T^{2} + 107666638 T^{4} - 546851229616 T^{6} + 2094008234358095 T^{8} - 6327905324283989088 T^{10} + \)\(15\!\cdots\!00\)\( T^{12} - 6327905324283989088 p^{6} T^{14} + 2094008234358095 p^{12} T^{16} - 546851229616 p^{18} T^{18} + 107666638 p^{24} T^{20} - 14448 p^{30} T^{22} + p^{36} T^{24} \) | |
| 19 | \( 1 + 12 T + 72 T^{2} + 71380 p T^{3} + 80627506 T^{4} + 1256173316 T^{5} + 928935243560 T^{6} + 122232243190932 T^{7} + 4866957296223951 T^{8} + 356984531389493592 T^{9} + 264815678913909840 p^{2} T^{10} + \)\(32\!\cdots\!16\)\( p T^{11} + \)\(12\!\cdots\!48\)\( T^{12} + \)\(32\!\cdots\!16\)\( p^{4} T^{13} + 264815678913909840 p^{8} T^{14} + 356984531389493592 p^{9} T^{15} + 4866957296223951 p^{12} T^{16} + 122232243190932 p^{15} T^{17} + 928935243560 p^{18} T^{18} + 1256173316 p^{21} T^{19} + 80627506 p^{24} T^{20} + 71380 p^{28} T^{21} + 72 p^{30} T^{22} + 12 p^{33} T^{23} + p^{36} T^{24} \) | |
| 23 | \( 1 - 12 p T + 17214 T^{2} + 1142692 T^{3} - 27820014 T^{4} - 28497242460 T^{5} + 1982704026150 T^{6} + 65861887634364 T^{7} + 99868618314741 p T^{8} + 875617922313069328 T^{9} - \)\(39\!\cdots\!16\)\( T^{10} - \)\(14\!\cdots\!48\)\( T^{11} + \)\(80\!\cdots\!56\)\( T^{12} - \)\(14\!\cdots\!48\)\( p^{3} T^{13} - \)\(39\!\cdots\!16\)\( p^{6} T^{14} + 875617922313069328 p^{9} T^{15} + 99868618314741 p^{13} T^{16} + 65861887634364 p^{15} T^{17} + 1982704026150 p^{18} T^{18} - 28497242460 p^{21} T^{19} - 27820014 p^{24} T^{20} + 1142692 p^{27} T^{21} + 17214 p^{30} T^{22} - 12 p^{34} T^{23} + p^{36} T^{24} \) | |
| 29 | \( 1 + 632 T + 193012 T^{2} + 31241744 T^{3} + 1162856136 T^{4} - 678887482016 T^{5} - 143704921685972 T^{6} - 1869446104826552 T^{7} + 4704974888482939551 T^{8} + \)\(10\!\cdots\!44\)\( T^{9} + \)\(72\!\cdots\!24\)\( T^{10} - \)\(12\!\cdots\!52\)\( T^{11} - \)\(36\!\cdots\!96\)\( T^{12} - \)\(12\!\cdots\!52\)\( p^{3} T^{13} + \)\(72\!\cdots\!24\)\( p^{6} T^{14} + \)\(10\!\cdots\!44\)\( p^{9} T^{15} + 4704974888482939551 p^{12} T^{16} - 1869446104826552 p^{15} T^{17} - 143704921685972 p^{18} T^{18} - 678887482016 p^{21} T^{19} + 1162856136 p^{24} T^{20} + 31241744 p^{27} T^{21} + 193012 p^{30} T^{22} + 632 p^{33} T^{23} + p^{36} T^{24} \) | |
| 31 | \( 1 - 188 T + 22934 T^{2} + 18386644 T^{3} - 3691984606 T^{4} + 776536429588 T^{5} + 99645628308078 T^{6} - 25120109404874988 T^{7} + 8682201249054683203 T^{8} - \)\(15\!\cdots\!80\)\( T^{9} + \)\(20\!\cdots\!44\)\( T^{10} + \)\(40\!\cdots\!64\)\( T^{11} - \)\(34\!\cdots\!20\)\( T^{12} + \)\(40\!\cdots\!64\)\( p^{3} T^{13} + \)\(20\!\cdots\!44\)\( p^{6} T^{14} - \)\(15\!\cdots\!80\)\( p^{9} T^{15} + 8682201249054683203 p^{12} T^{16} - 25120109404874988 p^{15} T^{17} + 99645628308078 p^{18} T^{18} + 776536429588 p^{21} T^{19} - 3691984606 p^{24} T^{20} + 18386644 p^{27} T^{21} + 22934 p^{30} T^{22} - 188 p^{33} T^{23} + p^{36} T^{24} \) | |
| 37 | \( 1 - 940 T + 400776 T^{2} - 69989748 T^{3} - 12837369392 T^{4} + 10952154351924 T^{5} - 2855667555891144 T^{6} + 200153198828812748 T^{7} + \)\(10\!\cdots\!87\)\( T^{8} - \)\(40\!\cdots\!08\)\( T^{9} + \)\(48\!\cdots\!68\)\( T^{10} + \)\(89\!\cdots\!52\)\( T^{11} - \)\(42\!\cdots\!88\)\( T^{12} + \)\(89\!\cdots\!52\)\( p^{3} T^{13} + \)\(48\!\cdots\!68\)\( p^{6} T^{14} - \)\(40\!\cdots\!08\)\( p^{9} T^{15} + \)\(10\!\cdots\!87\)\( p^{12} T^{16} + 200153198828812748 p^{15} T^{17} - 2855667555891144 p^{18} T^{18} + 10952154351924 p^{21} T^{19} - 12837369392 p^{24} T^{20} - 69989748 p^{27} T^{21} + 400776 p^{30} T^{22} - 940 p^{33} T^{23} + p^{36} T^{24} \) | |
| 41 | \( 1 + 176 T - 88034 T^{2} - 34173648 T^{3} + 827249858 T^{4} + 2621758340656 T^{5} + 280349260544702 T^{6} - 147031456973046352 T^{7} - 22298918693016177249 T^{8} + \)\(82\!\cdots\!96\)\( T^{9} + \)\(33\!\cdots\!96\)\( T^{10} - \)\(28\!\cdots\!24\)\( T^{11} - \)\(30\!\cdots\!88\)\( T^{12} - \)\(28\!\cdots\!24\)\( p^{3} T^{13} + \)\(33\!\cdots\!96\)\( p^{6} T^{14} + \)\(82\!\cdots\!96\)\( p^{9} T^{15} - 22298918693016177249 p^{12} T^{16} - 147031456973046352 p^{15} T^{17} + 280349260544702 p^{18} T^{18} + 2621758340656 p^{21} T^{19} + 827249858 p^{24} T^{20} - 34173648 p^{27} T^{21} - 88034 p^{30} T^{22} + 176 p^{33} T^{23} + p^{36} T^{24} \) | |
| 43 | \( 1 + 1360 T + 924800 T^{2} + 445088144 T^{3} + 187950043870 T^{4} + 76345837363280 T^{5} + 29065866207767168 T^{6} + 10133858719746215760 T^{7} + \)\(33\!\cdots\!95\)\( T^{8} + \)\(10\!\cdots\!56\)\( T^{9} + \)\(34\!\cdots\!40\)\( T^{10} + \)\(10\!\cdots\!60\)\( T^{11} + \)\(29\!\cdots\!16\)\( T^{12} + \)\(10\!\cdots\!60\)\( p^{3} T^{13} + \)\(34\!\cdots\!40\)\( p^{6} T^{14} + \)\(10\!\cdots\!56\)\( p^{9} T^{15} + \)\(33\!\cdots\!95\)\( p^{12} T^{16} + 10133858719746215760 p^{15} T^{17} + 29065866207767168 p^{18} T^{18} + 76345837363280 p^{21} T^{19} + 187950043870 p^{24} T^{20} + 445088144 p^{27} T^{21} + 924800 p^{30} T^{22} + 1360 p^{33} T^{23} + p^{36} T^{24} \) | |
| 47 | \( 1 - 580412 T^{2} + 182778426498 T^{4} - 40216713491295692 T^{6} + \)\(68\!\cdots\!79\)\( T^{8} - \)\(94\!\cdots\!00\)\( T^{10} + \)\(10\!\cdots\!68\)\( T^{12} - \)\(94\!\cdots\!00\)\( p^{6} T^{14} + \)\(68\!\cdots\!79\)\( p^{12} T^{16} - 40216713491295692 p^{18} T^{18} + 182778426498 p^{24} T^{20} - 580412 p^{30} T^{22} + p^{36} T^{24} \) | |
| 53 | \( 1 - 360 T + 64800 T^{2} - 135249864 T^{3} + 60322424078 T^{4} - 11373697219528 T^{5} + 9331900774784928 T^{6} - 6170118859401097768 T^{7} + \)\(17\!\cdots\!63\)\( T^{8} - \)\(52\!\cdots\!24\)\( T^{9} + \)\(41\!\cdots\!00\)\( T^{10} - \)\(14\!\cdots\!20\)\( T^{11} + \)\(24\!\cdots\!08\)\( T^{12} - \)\(14\!\cdots\!20\)\( p^{3} T^{13} + \)\(41\!\cdots\!00\)\( p^{6} T^{14} - \)\(52\!\cdots\!24\)\( p^{9} T^{15} + \)\(17\!\cdots\!63\)\( p^{12} T^{16} - 6170118859401097768 p^{15} T^{17} + 9331900774784928 p^{18} T^{18} - 11373697219528 p^{21} T^{19} + 60322424078 p^{24} T^{20} - 135249864 p^{27} T^{21} + 64800 p^{30} T^{22} - 360 p^{33} T^{23} + p^{36} T^{24} \) | |
| 59 | \( 1 - 584 T + 170528 T^{2} + 175550056 T^{3} - 94146703170 T^{4} - 18580823027848 T^{5} + 42314760727238560 T^{6} - 18131962503470660184 T^{7} - \)\(19\!\cdots\!73\)\( T^{8} + \)\(38\!\cdots\!88\)\( T^{9} - \)\(96\!\cdots\!00\)\( T^{10} - \)\(60\!\cdots\!68\)\( T^{11} + \)\(52\!\cdots\!08\)\( T^{12} - \)\(60\!\cdots\!68\)\( p^{3} T^{13} - \)\(96\!\cdots\!00\)\( p^{6} T^{14} + \)\(38\!\cdots\!88\)\( p^{9} T^{15} - \)\(19\!\cdots\!73\)\( p^{12} T^{16} - 18131962503470660184 p^{15} T^{17} + 42314760727238560 p^{18} T^{18} - 18580823027848 p^{21} T^{19} - 94146703170 p^{24} T^{20} + 175550056 p^{27} T^{21} + 170528 p^{30} T^{22} - 584 p^{33} T^{23} + p^{36} T^{24} \) | |
| 61 | \( 1 + 1052 T + 1006376 T^{2} + 412979636 T^{3} + 133288994960 T^{4} - 72852168865956 T^{5} - 77682489128187272 T^{6} - 64250295850085738220 T^{7} - \)\(24\!\cdots\!17\)\( T^{8} - \)\(69\!\cdots\!76\)\( T^{9} + \)\(27\!\cdots\!44\)\( T^{10} + \)\(28\!\cdots\!40\)\( T^{11} + \)\(20\!\cdots\!96\)\( T^{12} + \)\(28\!\cdots\!40\)\( p^{3} T^{13} + \)\(27\!\cdots\!44\)\( p^{6} T^{14} - \)\(69\!\cdots\!76\)\( p^{9} T^{15} - \)\(24\!\cdots\!17\)\( p^{12} T^{16} - 64250295850085738220 p^{15} T^{17} - 77682489128187272 p^{18} T^{18} - 72852168865956 p^{21} T^{19} + 133288994960 p^{24} T^{20} + 412979636 p^{27} T^{21} + 1006376 p^{30} T^{22} + 1052 p^{33} T^{23} + p^{36} T^{24} \) | |
| 67 | \( ( 1 - 540 T + 1237196 T^{2} - 437516948 T^{3} + 687431155711 T^{4} - 191267075797288 T^{5} + 250196846973981368 T^{6} - 191267075797288 p^{3} T^{7} + 687431155711 p^{6} T^{8} - 437516948 p^{9} T^{9} + 1237196 p^{12} T^{10} - 540 p^{15} T^{11} + p^{18} T^{12} )^{2} \) | |
| 71 | \( 1 + 28 T - 195882 T^{2} - 194007508 T^{3} + 13906393890 T^{4} + 24274230169564 T^{5} + 39419381006302414 T^{6} - 24734369575425164292 T^{7} - \)\(95\!\cdots\!37\)\( T^{8} + \)\(58\!\cdots\!96\)\( p T^{9} + \)\(42\!\cdots\!20\)\( T^{10} - \)\(13\!\cdots\!48\)\( T^{11} - \)\(63\!\cdots\!80\)\( T^{12} - \)\(13\!\cdots\!48\)\( p^{3} T^{13} + \)\(42\!\cdots\!20\)\( p^{6} T^{14} + \)\(58\!\cdots\!96\)\( p^{10} T^{15} - \)\(95\!\cdots\!37\)\( p^{12} T^{16} - 24734369575425164292 p^{15} T^{17} + 39419381006302414 p^{18} T^{18} + 24274230169564 p^{21} T^{19} + 13906393890 p^{24} T^{20} - 194007508 p^{27} T^{21} - 195882 p^{30} T^{22} + 28 p^{33} T^{23} + p^{36} T^{24} \) | |
| 73 | \( 1 - 824 T + 588222 T^{2} + 9262472 T^{3} - 118766234814 T^{4} + 230361433132360 T^{5} - 186489223297230882 T^{6} + \)\(15\!\cdots\!92\)\( T^{7} - \)\(70\!\cdots\!77\)\( T^{8} + \)\(10\!\cdots\!48\)\( T^{9} + \)\(24\!\cdots\!96\)\( T^{10} - \)\(36\!\cdots\!68\)\( p T^{11} + \)\(31\!\cdots\!92\)\( p^{2} T^{12} - \)\(36\!\cdots\!68\)\( p^{4} T^{13} + \)\(24\!\cdots\!96\)\( p^{6} T^{14} + \)\(10\!\cdots\!48\)\( p^{9} T^{15} - \)\(70\!\cdots\!77\)\( p^{12} T^{16} + \)\(15\!\cdots\!92\)\( p^{15} T^{17} - 186489223297230882 p^{18} T^{18} + 230361433132360 p^{21} T^{19} - 118766234814 p^{24} T^{20} + 9262472 p^{27} T^{21} + 588222 p^{30} T^{22} - 824 p^{33} T^{23} + p^{36} T^{24} \) | |
| 79 | \( 1 + 196 T + 788046 T^{2} + 351355900 T^{3} + 349469377586 T^{4} + 376015608150652 T^{5} + 21556020940992630 T^{6} + \)\(18\!\cdots\!52\)\( T^{7} + \)\(20\!\cdots\!43\)\( T^{8} + \)\(54\!\cdots\!52\)\( T^{9} + \)\(53\!\cdots\!64\)\( T^{10} - \)\(49\!\cdots\!12\)\( T^{11} + \)\(41\!\cdots\!72\)\( T^{12} - \)\(49\!\cdots\!12\)\( p^{3} T^{13} + \)\(53\!\cdots\!64\)\( p^{6} T^{14} + \)\(54\!\cdots\!52\)\( p^{9} T^{15} + \)\(20\!\cdots\!43\)\( p^{12} T^{16} + \)\(18\!\cdots\!52\)\( p^{15} T^{17} + 21556020940992630 p^{18} T^{18} + 376015608150652 p^{21} T^{19} + 349469377586 p^{24} T^{20} + 351355900 p^{27} T^{21} + 788046 p^{30} T^{22} + 196 p^{33} T^{23} + p^{36} T^{24} \) | |
| 83 | \( 1 - 1008 T + 508032 T^{2} - 608566144 T^{3} + 1311140396574 T^{4} - 925961362981888 T^{5} + 452444151744975104 T^{6} - \)\(51\!\cdots\!40\)\( T^{7} + \)\(71\!\cdots\!79\)\( T^{8} - \)\(40\!\cdots\!08\)\( T^{9} + \)\(20\!\cdots\!72\)\( T^{10} - \)\(22\!\cdots\!60\)\( T^{11} + \)\(25\!\cdots\!56\)\( T^{12} - \)\(22\!\cdots\!60\)\( p^{3} T^{13} + \)\(20\!\cdots\!72\)\( p^{6} T^{14} - \)\(40\!\cdots\!08\)\( p^{9} T^{15} + \)\(71\!\cdots\!79\)\( p^{12} T^{16} - \)\(51\!\cdots\!40\)\( p^{15} T^{17} + 452444151744975104 p^{18} T^{18} - 925961362981888 p^{21} T^{19} + 1311140396574 p^{24} T^{20} - 608566144 p^{27} T^{21} + 508032 p^{30} T^{22} - 1008 p^{33} T^{23} + p^{36} T^{24} \) | |
| 89 | \( 1 - 5719348 T^{2} + 16180825465546 T^{4} - 29845545554168255940 T^{6} + \)\(39\!\cdots\!99\)\( T^{8} - \)\(40\!\cdots\!04\)\( T^{10} + \)\(32\!\cdots\!60\)\( T^{12} - \)\(40\!\cdots\!04\)\( p^{6} T^{14} + \)\(39\!\cdots\!99\)\( p^{12} T^{16} - 29845545554168255940 p^{18} T^{18} + 16180825465546 p^{24} T^{20} - 5719348 p^{30} T^{22} + p^{36} T^{24} \) | |
| 97 | \( 1 + 904 T + 307166 T^{2} - 2157443544 T^{3} - 1987213960190 T^{4} - 448060583775704 T^{5} + 3336323390652175454 T^{6} + \)\(24\!\cdots\!08\)\( T^{7} + \)\(10\!\cdots\!15\)\( T^{8} - \)\(37\!\cdots\!80\)\( T^{9} - \)\(19\!\cdots\!56\)\( T^{10} + \)\(70\!\cdots\!36\)\( T^{11} + \)\(35\!\cdots\!84\)\( T^{12} + \)\(70\!\cdots\!36\)\( p^{3} T^{13} - \)\(19\!\cdots\!56\)\( p^{6} T^{14} - \)\(37\!\cdots\!80\)\( p^{9} T^{15} + \)\(10\!\cdots\!15\)\( p^{12} T^{16} + \)\(24\!\cdots\!08\)\( p^{15} T^{17} + 3336323390652175454 p^{18} T^{18} - 448060583775704 p^{21} T^{19} - 1987213960190 p^{24} T^{20} - 2157443544 p^{27} T^{21} + 307166 p^{30} T^{22} + 904 p^{33} T^{23} + p^{36} T^{24} \) | |
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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
−4.02979697545569547495695498347, −4.00023650876041296610517886531, −3.72051608316568961189244467244, −3.68207029793796593682328201666, −3.65296714689534970014997696915, −3.44275655865975331089474925115, −3.16219143316689699283564401750, −3.15102900144475810889797785895, −2.85417761610707792527095892233, −2.83871545906268546927564944053, −2.79490022314288842611266018592, −2.70251401933635351336898339641, −2.46415908031619556691069470176, −2.19611943799874642721036130634, −2.08318463217322256718135145331, −2.02849514457069039112720814053, −1.93480625798066062444964799574, −1.52384047431185988825626213261, −1.48258584488545648760520471519, −1.31496816022601126565227364282, −1.29551173051216310299966057325, −0.75680508762288307582152650700, −0.63217503814916052706373984990, −0.42464322717404608287821627834, −0.01059253751621469844543026243, 0.01059253751621469844543026243, 0.42464322717404608287821627834, 0.63217503814916052706373984990, 0.75680508762288307582152650700, 1.29551173051216310299966057325, 1.31496816022601126565227364282, 1.48258584488545648760520471519, 1.52384047431185988825626213261, 1.93480625798066062444964799574, 2.02849514457069039112720814053, 2.08318463217322256718135145331, 2.19611943799874642721036130634, 2.46415908031619556691069470176, 2.70251401933635351336898339641, 2.79490022314288842611266018592, 2.83871545906268546927564944053, 2.85417761610707792527095892233, 3.15102900144475810889797785895, 3.16219143316689699283564401750, 3.44275655865975331089474925115, 3.65296714689534970014997696915, 3.68207029793796593682328201666, 3.72051608316568961189244467244, 4.00023650876041296610517886531, 4.02979697545569547495695498347
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.