Dirichlet series
| L(s) = 1 | − 4·2-s − 4·3-s + 8·4-s − 20·5-s + 16·6-s − 4·7-s − 12·8-s − 24·9-s + 80·10-s + 40·11-s − 32·12-s + 16·14-s + 80·15-s + 94·16-s + 52·17-s + 96·18-s − 12·19-s − 160·20-s + 16·21-s − 160·22-s − 276·23-s + 48·24-s − 32·25-s − 68·27-s − 32·28-s + 632·29-s − 320·30-s + ⋯ |
| L(s) = 1 | − 1.41·2-s − 0.769·3-s + 4-s − 1.78·5-s + 1.08·6-s − 0.215·7-s − 0.530·8-s − 8/9·9-s + 2.52·10-s + 1.09·11-s − 0.769·12-s + 0.305·14-s + 1.37·15-s + 1.46·16-s + 0.741·17-s + 1.25·18-s − 0.144·19-s − 1.78·20-s + 0.166·21-s − 1.55·22-s − 2.50·23-s + 0.408·24-s − 0.255·25-s − 0.484·27-s − 0.215·28-s + 4.04·29-s − 1.94·30-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(17^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(17^{12}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(24\) |
| Conductor: | \(17^{12}\) |
| Sign: | $1$ |
| Analytic conductor: | \(1.03700\) |
| Root analytic conductor: | \(1.00151\) |
| Motivic weight: | \(3\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((24,\ 17^{12} ,\ ( \ : [3/2]^{12} ),\ 1 )\) |
Particular Values
| \(L(2)\) | \(\approx\) | \(0.2080528806\) |
| \(L(\frac12)\) | \(\approx\) | \(0.2080528806\) |
| \(L(\frac{5}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 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 | \( 1 + p^{2} T + p^{3} T^{2} + 3 p^{2} T^{3} - 31 p T^{4} - 97 p^{2} T^{5} - 123 p^{3} T^{6} - 925 p^{2} T^{7} - 6783 T^{8} + 815 p^{3} T^{9} + 1569 p^{5} T^{10} + 1649 p^{7} T^{11} + 13335 p^{6} T^{12} + 1649 p^{10} T^{13} + 1569 p^{11} T^{14} + 815 p^{12} T^{15} - 6783 p^{12} T^{16} - 925 p^{17} T^{17} - 123 p^{21} T^{18} - 97 p^{23} T^{19} - 31 p^{25} T^{20} + 3 p^{29} T^{21} + p^{33} T^{22} + p^{35} T^{23} + p^{36} T^{24} \) |
| 3 | \( 1 + 4 T + 40 T^{2} + 4 p^{4} T^{3} + 1520 T^{4} + 5668 p T^{5} + 10352 p^{2} T^{6} + 207940 p T^{7} + 3542815 T^{8} + 19106488 T^{9} + 116779528 T^{10} + 203839592 p T^{11} + 3634702400 T^{12} + 203839592 p^{4} T^{13} + 116779528 p^{6} T^{14} + 19106488 p^{9} T^{15} + 3542815 p^{12} T^{16} + 207940 p^{16} T^{17} + 10352 p^{20} T^{18} + 5668 p^{22} T^{19} + 1520 p^{24} T^{20} + 4 p^{31} T^{21} + 40 p^{30} T^{22} + 4 p^{33} T^{23} + p^{36} T^{24} \) | |
| 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
−7.54642300684526191389181272974, −7.07980552522260115628854611252, −6.99999897075463800161405171289, −6.87717063410500038810814649426, −6.69215851517051371857735440021, −6.52557220776585370744090225359, −6.17943458880210868926969019175, −6.07273228724659865446232937980, −6.06269400590765625480868872416, −5.99428858372868210881898298167, −5.69818131037012740520433512186, −5.50055939819920528126720465314, −5.19602822545562950495250032332, −4.64922345250640003947191320486, −4.62884197489022035717104144423, −4.58067247302031778353835872755, −4.38480225524344062313562012776, −3.83434324346232986494305054609, −3.79242950438097635350486317070, −3.49237573874081310895319304065, −3.18412007922491424740425045963, −2.80811572346532257613209573940, −2.48354463017318552863143116580, −1.59437853974255174968076439210, −0.62705736002593820342591753138, 0.62705736002593820342591753138, 1.59437853974255174968076439210, 2.48354463017318552863143116580, 2.80811572346532257613209573940, 3.18412007922491424740425045963, 3.49237573874081310895319304065, 3.79242950438097635350486317070, 3.83434324346232986494305054609, 4.38480225524344062313562012776, 4.58067247302031778353835872755, 4.62884197489022035717104144423, 4.64922345250640003947191320486, 5.19602822545562950495250032332, 5.50055939819920528126720465314, 5.69818131037012740520433512186, 5.99428858372868210881898298167, 6.06269400590765625480868872416, 6.07273228724659865446232937980, 6.17943458880210868926969019175, 6.52557220776585370744090225359, 6.69215851517051371857735440021, 6.87717063410500038810814649426, 6.99999897075463800161405171289, 7.07980552522260115628854611252, 7.54642300684526191389181272974
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.