Dirichlet series
| L(s) = 1 | + 10·2-s + 3·3-s + 32·4-s + 29·5-s + 30·6-s + 39·7-s − 98·9-s + 290·10-s + 132·11-s + 96·12-s + 171·13-s + 390·14-s + 87·15-s − 219·16-s + 204·17-s − 980·18-s − 3·19-s + 928·20-s + 117·21-s + 1.32e3·22-s + 268·23-s − 132·25-s + 1.71e3·26-s − 432·27-s + 1.24e3·28-s + 466·29-s + 870·30-s + ⋯ |
| L(s) = 1 | + 3.53·2-s + 0.577·3-s + 4·4-s + 2.59·5-s + 2.04·6-s + 2.10·7-s − 3.62·9-s + 9.17·10-s + 3.61·11-s + 2.30·12-s + 3.64·13-s + 7.44·14-s + 1.49·15-s − 3.42·16-s + 2.91·17-s − 12.8·18-s − 0.0362·19-s + 10.3·20-s + 1.21·21-s + 12.7·22-s + 2.42·23-s − 1.05·25-s + 12.8·26-s − 3.07·27-s + 8.42·28-s + 2.98·29-s + 5.29·30-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(11^{12} \cdot 17^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(11^{12} \cdot 17^{12}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(24\) |
| Conductor: | \(11^{12} \cdot 17^{12}\) |
| Sign: | $1$ |
| Analytic conductor: | \(3.25455\times 10^{12}\) |
| Root analytic conductor: | \(3.32164\) |
| Motivic weight: | \(3\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((24,\ 11^{12} \cdot 17^{12} ,\ ( \ : [3/2]^{12} ),\ 1 )\) |
Particular Values
| \(L(2)\) | \(\approx\) | \(1184.737108\) |
| \(L(\frac12)\) | \(\approx\) | \(1184.737108\) |
| \(L(\frac{5}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 11 | \( ( 1 - p T )^{12} \) |
| 17 | \( ( 1 - p T )^{12} \) | |
| good | 2 | \( 1 - 5 p T + 17 p^{2} T^{2} - 45 p^{3} T^{3} + 1643 T^{4} - 1697 p^{2} T^{5} + 6583 p^{2} T^{6} - 48331 p T^{7} + 84453 p^{2} T^{8} - 279061 p^{2} T^{9} + 218849 p^{4} T^{10} - 327529 p^{5} T^{11} + 1887605 p^{4} T^{12} - 327529 p^{8} T^{13} + 218849 p^{10} T^{14} - 279061 p^{11} T^{15} + 84453 p^{14} T^{16} - 48331 p^{16} T^{17} + 6583 p^{20} T^{18} - 1697 p^{23} T^{19} + 1643 p^{24} T^{20} - 45 p^{30} T^{21} + 17 p^{32} T^{22} - 5 p^{34} T^{23} + p^{36} T^{24} \) |
| 3 | \( 1 - p T + 107 T^{2} - 61 p T^{3} + 2071 p T^{4} - 143 T^{5} + 254812 T^{6} + 98335 p T^{7} + 9472471 T^{8} + 1486129 p^{2} T^{9} + 112480655 p T^{10} + 340006457 T^{11} + 10143163294 T^{12} + 340006457 p^{3} T^{13} + 112480655 p^{7} T^{14} + 1486129 p^{11} T^{15} + 9472471 p^{12} T^{16} + 98335 p^{16} T^{17} + 254812 p^{18} T^{18} - 143 p^{21} T^{19} + 2071 p^{25} T^{20} - 61 p^{28} T^{21} + 107 p^{30} T^{22} - p^{34} T^{23} + p^{36} T^{24} \) | |
| 5 | \( 1 - 29 T + 973 T^{2} - 4003 p T^{3} + 429697 T^{4} - 1473889 p T^{5} + 126424214 T^{6} - 1902387831 T^{7} + 222553087 p^{3} T^{8} - 373429226873 T^{9} + 4803649397643 T^{10} - 58028081231023 T^{11} + 667366992196162 T^{12} - 58028081231023 p^{3} T^{13} + 4803649397643 p^{6} T^{14} - 373429226873 p^{9} T^{15} + 222553087 p^{15} T^{16} - 1902387831 p^{15} T^{17} + 126424214 p^{18} T^{18} - 1473889 p^{22} T^{19} + 429697 p^{24} T^{20} - 4003 p^{28} T^{21} + 973 p^{30} T^{22} - 29 p^{33} T^{23} + p^{36} T^{24} \) | |
| 7 | \( 1 - 39 T + 2603 T^{2} - 84979 T^{3} + 3459222 T^{4} - 96598288 T^{5} + 3008875495 T^{6} - 73396518476 T^{7} + 1911886320559 T^{8} - 41226312406809 T^{9} + 933007460109086 T^{10} - 17919582958395793 T^{11} + 359156908188652724 T^{12} - 17919582958395793 p^{3} T^{13} + 933007460109086 p^{6} T^{14} - 41226312406809 p^{9} T^{15} + 1911886320559 p^{12} T^{16} - 73396518476 p^{15} T^{17} + 3008875495 p^{18} T^{18} - 96598288 p^{21} T^{19} + 3459222 p^{24} T^{20} - 84979 p^{27} T^{21} + 2603 p^{30} T^{22} - 39 p^{33} T^{23} + p^{36} T^{24} \) | |
| 13 | \( 1 - 171 T + 154 p^{2} T^{2} - 2632158 T^{3} + 236692142 T^{4} - 17467042902 T^{5} + 1170520591646 T^{6} - 70025833519899 T^{7} + 3898330446005975 T^{8} - 204577494288672946 T^{9} + 786632462071920848 p T^{10} - \)\(49\!\cdots\!16\)\( T^{11} + \)\(23\!\cdots\!08\)\( T^{12} - \)\(49\!\cdots\!16\)\( p^{3} T^{13} + 786632462071920848 p^{7} T^{14} - 204577494288672946 p^{9} T^{15} + 3898330446005975 p^{12} T^{16} - 70025833519899 p^{15} T^{17} + 1170520591646 p^{18} T^{18} - 17467042902 p^{21} T^{19} + 236692142 p^{24} T^{20} - 2632158 p^{27} T^{21} + 154 p^{32} T^{22} - 171 p^{33} T^{23} + p^{36} T^{24} \) | |
| 19 | \( 1 + 3 T + 2556 p T^{2} - 125488 T^{3} + 1183347984 T^{4} - 6875428980 T^{5} + 19252505049784 T^{6} - 138733477519311 T^{7} + 233229565569517447 T^{8} - 1746438462349099850 T^{9} + \)\(22\!\cdots\!00\)\( T^{10} - \)\(15\!\cdots\!32\)\( T^{11} + \)\(16\!\cdots\!64\)\( T^{12} - \)\(15\!\cdots\!32\)\( p^{3} T^{13} + \)\(22\!\cdots\!00\)\( p^{6} T^{14} - 1746438462349099850 p^{9} T^{15} + 233229565569517447 p^{12} T^{16} - 138733477519311 p^{15} T^{17} + 19252505049784 p^{18} T^{18} - 6875428980 p^{21} T^{19} + 1183347984 p^{24} T^{20} - 125488 p^{27} T^{21} + 2556 p^{31} T^{22} + 3 p^{33} T^{23} + p^{36} T^{24} \) | |
| 23 | \( 1 - 268 T + 91845 T^{2} - 16319344 T^{3} + 3207857107 T^{4} - 407992694236 T^{5} + 55820991947104 T^{6} - 4629097961311932 T^{7} + 394286896646364337 T^{8} + 4197360746850904800 T^{9} - \)\(13\!\cdots\!91\)\( p T^{10} + \)\(86\!\cdots\!88\)\( T^{11} - \)\(39\!\cdots\!74\)\( p T^{12} + \)\(86\!\cdots\!88\)\( p^{3} T^{13} - \)\(13\!\cdots\!91\)\( p^{7} T^{14} + 4197360746850904800 p^{9} T^{15} + 394286896646364337 p^{12} T^{16} - 4629097961311932 p^{15} T^{17} + 55820991947104 p^{18} T^{18} - 407992694236 p^{21} T^{19} + 3207857107 p^{24} T^{20} - 16319344 p^{27} T^{21} + 91845 p^{30} T^{22} - 268 p^{33} T^{23} + p^{36} T^{24} \) | |
| 29 | \( 1 - 466 T + 245821 T^{2} - 81938374 T^{3} + 27063680436 T^{4} - 7256401584118 T^{5} + 1866262738658797 T^{6} - 422197622900960258 T^{7} + 90760744384459130483 T^{8} - \)\(17\!\cdots\!20\)\( T^{9} + \)\(32\!\cdots\!82\)\( T^{10} - \)\(56\!\cdots\!96\)\( T^{11} + \)\(91\!\cdots\!92\)\( T^{12} - \)\(56\!\cdots\!96\)\( p^{3} T^{13} + \)\(32\!\cdots\!82\)\( p^{6} T^{14} - \)\(17\!\cdots\!20\)\( p^{9} T^{15} + 90760744384459130483 p^{12} T^{16} - 422197622900960258 p^{15} T^{17} + 1866262738658797 p^{18} T^{18} - 7256401584118 p^{21} T^{19} + 27063680436 p^{24} T^{20} - 81938374 p^{27} T^{21} + 245821 p^{30} T^{22} - 466 p^{33} T^{23} + p^{36} T^{24} \) | |
| 31 | \( 1 - 98 T + 294661 T^{2} - 24718928 T^{3} + 41009518871 T^{4} - 2958996271974 T^{5} + 3575267895018540 T^{6} - 222985432782127510 T^{7} + \)\(21\!\cdots\!61\)\( T^{8} - \)\(11\!\cdots\!64\)\( T^{9} + \)\(98\!\cdots\!71\)\( T^{10} - \)\(46\!\cdots\!18\)\( T^{11} + \)\(33\!\cdots\!90\)\( T^{12} - \)\(46\!\cdots\!18\)\( p^{3} T^{13} + \)\(98\!\cdots\!71\)\( p^{6} T^{14} - \)\(11\!\cdots\!64\)\( p^{9} T^{15} + \)\(21\!\cdots\!61\)\( p^{12} T^{16} - 222985432782127510 p^{15} T^{17} + 3575267895018540 p^{18} T^{18} - 2958996271974 p^{21} T^{19} + 41009518871 p^{24} T^{20} - 24718928 p^{27} T^{21} + 294661 p^{30} T^{22} - 98 p^{33} T^{23} + p^{36} T^{24} \) | |
| 37 | \( 1 - 201 T + 319831 T^{2} - 33955883 T^{3} + 46926516819 T^{4} - 1851115658881 T^{5} + 4729756373718742 T^{6} + 156247450384237 p T^{7} + \)\(38\!\cdots\!73\)\( T^{8} + \)\(82\!\cdots\!63\)\( T^{9} + \)\(25\!\cdots\!09\)\( T^{10} + \)\(77\!\cdots\!77\)\( T^{11} + \)\(14\!\cdots\!38\)\( T^{12} + \)\(77\!\cdots\!77\)\( p^{3} T^{13} + \)\(25\!\cdots\!09\)\( p^{6} T^{14} + \)\(82\!\cdots\!63\)\( p^{9} T^{15} + \)\(38\!\cdots\!73\)\( p^{12} T^{16} + 156247450384237 p^{16} T^{17} + 4729756373718742 p^{18} T^{18} - 1851115658881 p^{21} T^{19} + 46926516819 p^{24} T^{20} - 33955883 p^{27} T^{21} + 319831 p^{30} T^{22} - 201 p^{33} T^{23} + p^{36} T^{24} \) | |
| 41 | \( 1 - 221 T + 566007 T^{2} - 134407275 T^{3} + 157565790364 T^{4} - 37737419348812 T^{5} + 28586831453049705 T^{6} - 6584717945859303812 T^{7} + \)\(37\!\cdots\!59\)\( T^{8} - \)\(80\!\cdots\!27\)\( T^{9} + \)\(37\!\cdots\!28\)\( T^{10} - \)\(72\!\cdots\!53\)\( T^{11} + \)\(29\!\cdots\!76\)\( T^{12} - \)\(72\!\cdots\!53\)\( p^{3} T^{13} + \)\(37\!\cdots\!28\)\( p^{6} T^{14} - \)\(80\!\cdots\!27\)\( p^{9} T^{15} + \)\(37\!\cdots\!59\)\( p^{12} T^{16} - 6584717945859303812 p^{15} T^{17} + 28586831453049705 p^{18} T^{18} - 37737419348812 p^{21} T^{19} + 157565790364 p^{24} T^{20} - 134407275 p^{27} T^{21} + 566007 p^{30} T^{22} - 221 p^{33} T^{23} + p^{36} T^{24} \) | |
| 43 | \( 1 - 707 T + 691830 T^{2} - 328522812 T^{3} + 190739914158 T^{4} - 67741235284348 T^{5} + 29523350181235934 T^{6} - 8156309947651455109 T^{7} + \)\(29\!\cdots\!63\)\( T^{8} - \)\(66\!\cdots\!10\)\( T^{9} + \)\(23\!\cdots\!92\)\( T^{10} - \)\(44\!\cdots\!68\)\( T^{11} + \)\(17\!\cdots\!76\)\( T^{12} - \)\(44\!\cdots\!68\)\( p^{3} T^{13} + \)\(23\!\cdots\!92\)\( p^{6} T^{14} - \)\(66\!\cdots\!10\)\( p^{9} T^{15} + \)\(29\!\cdots\!63\)\( p^{12} T^{16} - 8156309947651455109 p^{15} T^{17} + 29523350181235934 p^{18} T^{18} - 67741235284348 p^{21} T^{19} + 190739914158 p^{24} T^{20} - 328522812 p^{27} T^{21} + 691830 p^{30} T^{22} - 707 p^{33} T^{23} + p^{36} T^{24} \) | |
| 47 | \( 1 + 79 T + 721857 T^{2} + 52337003 T^{3} + 246101003072 T^{4} + 21539206405752 T^{5} + 53364459055336893 T^{6} + 6404537304103893012 T^{7} + \)\(84\!\cdots\!63\)\( T^{8} + \)\(13\!\cdots\!09\)\( T^{9} + \)\(10\!\cdots\!90\)\( T^{10} + \)\(19\!\cdots\!53\)\( T^{11} + \)\(11\!\cdots\!04\)\( T^{12} + \)\(19\!\cdots\!53\)\( p^{3} T^{13} + \)\(10\!\cdots\!90\)\( p^{6} T^{14} + \)\(13\!\cdots\!09\)\( p^{9} T^{15} + \)\(84\!\cdots\!63\)\( p^{12} T^{16} + 6404537304103893012 p^{15} T^{17} + 53364459055336893 p^{18} T^{18} + 21539206405752 p^{21} T^{19} + 246101003072 p^{24} T^{20} + 52337003 p^{27} T^{21} + 721857 p^{30} T^{22} + 79 p^{33} T^{23} + p^{36} T^{24} \) | |
| 53 | \( 1 - 932 T + 1390741 T^{2} - 1043363398 T^{3} + 907704498984 T^{4} - 574185698545822 T^{5} + 375247440401731177 T^{6} - \)\(20\!\cdots\!88\)\( T^{7} + \)\(11\!\cdots\!91\)\( T^{8} - \)\(52\!\cdots\!84\)\( T^{9} + \)\(24\!\cdots\!50\)\( T^{10} - \)\(10\!\cdots\!20\)\( T^{11} + \)\(41\!\cdots\!00\)\( T^{12} - \)\(10\!\cdots\!20\)\( p^{3} T^{13} + \)\(24\!\cdots\!50\)\( p^{6} T^{14} - \)\(52\!\cdots\!84\)\( p^{9} T^{15} + \)\(11\!\cdots\!91\)\( p^{12} T^{16} - \)\(20\!\cdots\!88\)\( p^{15} T^{17} + 375247440401731177 p^{18} T^{18} - 574185698545822 p^{21} T^{19} + 907704498984 p^{24} T^{20} - 1043363398 p^{27} T^{21} + 1390741 p^{30} T^{22} - 932 p^{33} T^{23} + p^{36} T^{24} \) | |
| 59 | \( 1 + 72 T + 775012 T^{2} + 34356784 T^{3} + 340236131248 T^{4} + 25524604476880 T^{5} + 116097339600791504 T^{6} + 11234446532267656168 T^{7} + \)\(32\!\cdots\!16\)\( T^{8} + \)\(28\!\cdots\!88\)\( T^{9} + \)\(82\!\cdots\!40\)\( T^{10} + \)\(62\!\cdots\!68\)\( T^{11} + \)\(18\!\cdots\!86\)\( T^{12} + \)\(62\!\cdots\!68\)\( p^{3} T^{13} + \)\(82\!\cdots\!40\)\( p^{6} T^{14} + \)\(28\!\cdots\!88\)\( p^{9} T^{15} + \)\(32\!\cdots\!16\)\( p^{12} T^{16} + 11234446532267656168 p^{15} T^{17} + 116097339600791504 p^{18} T^{18} + 25524604476880 p^{21} T^{19} + 340236131248 p^{24} T^{20} + 34356784 p^{27} T^{21} + 775012 p^{30} T^{22} + 72 p^{33} T^{23} + p^{36} T^{24} \) | |
| 61 | \( 1 + 1012 T + 1806712 T^{2} + 1189159828 T^{3} + 1266103747962 T^{4} + 582992152341524 T^{5} + 490775528409723928 T^{6} + \)\(14\!\cdots\!28\)\( T^{7} + \)\(11\!\cdots\!59\)\( T^{8} + \)\(13\!\cdots\!96\)\( T^{9} + \)\(19\!\cdots\!48\)\( T^{10} - \)\(26\!\cdots\!12\)\( T^{11} + \)\(31\!\cdots\!44\)\( T^{12} - \)\(26\!\cdots\!12\)\( p^{3} T^{13} + \)\(19\!\cdots\!48\)\( p^{6} T^{14} + \)\(13\!\cdots\!96\)\( p^{9} T^{15} + \)\(11\!\cdots\!59\)\( p^{12} T^{16} + \)\(14\!\cdots\!28\)\( p^{15} T^{17} + 490775528409723928 p^{18} T^{18} + 582992152341524 p^{21} T^{19} + 1266103747962 p^{24} T^{20} + 1189159828 p^{27} T^{21} + 1806712 p^{30} T^{22} + 1012 p^{33} T^{23} + p^{36} T^{24} \) | |
| 67 | \( 1 + 598 T + 2339280 T^{2} + 1438706050 T^{3} + 2723629759880 T^{4} + 1660666659288790 T^{5} + 2082154467182482028 T^{6} + \)\(12\!\cdots\!10\)\( T^{7} + \)\(11\!\cdots\!92\)\( T^{8} + \)\(63\!\cdots\!14\)\( T^{9} + \)\(49\!\cdots\!32\)\( T^{10} + \)\(25\!\cdots\!06\)\( T^{11} + \)\(16\!\cdots\!58\)\( T^{12} + \)\(25\!\cdots\!06\)\( p^{3} T^{13} + \)\(49\!\cdots\!32\)\( p^{6} T^{14} + \)\(63\!\cdots\!14\)\( p^{9} T^{15} + \)\(11\!\cdots\!92\)\( p^{12} T^{16} + \)\(12\!\cdots\!10\)\( p^{15} T^{17} + 2082154467182482028 p^{18} T^{18} + 1660666659288790 p^{21} T^{19} + 2723629759880 p^{24} T^{20} + 1438706050 p^{27} T^{21} + 2339280 p^{30} T^{22} + 598 p^{33} T^{23} + p^{36} T^{24} \) | |
| 71 | \( 1 - 698 T + 2584189 T^{2} - 1951291452 T^{3} + 3558399541387 T^{4} - 2553952089295106 T^{5} + 3306221054992996428 T^{6} - \)\(21\!\cdots\!06\)\( T^{7} + \)\(22\!\cdots\!01\)\( T^{8} - \)\(13\!\cdots\!00\)\( T^{9} + \)\(11\!\cdots\!19\)\( T^{10} - \)\(61\!\cdots\!02\)\( T^{11} + \)\(46\!\cdots\!34\)\( T^{12} - \)\(61\!\cdots\!02\)\( p^{3} T^{13} + \)\(11\!\cdots\!19\)\( p^{6} T^{14} - \)\(13\!\cdots\!00\)\( p^{9} T^{15} + \)\(22\!\cdots\!01\)\( p^{12} T^{16} - \)\(21\!\cdots\!06\)\( p^{15} T^{17} + 3306221054992996428 p^{18} T^{18} - 2553952089295106 p^{21} T^{19} + 3558399541387 p^{24} T^{20} - 1951291452 p^{27} T^{21} + 2584189 p^{30} T^{22} - 698 p^{33} T^{23} + p^{36} T^{24} \) | |
| 73 | \( 1 - 195 T + 1433195 T^{2} + 66383959 T^{3} + 1246496536554 T^{4} + 91665334499364 T^{5} + 903018661516769731 T^{6} + 63959311909282630772 T^{7} + \)\(51\!\cdots\!27\)\( T^{8} + \)\(42\!\cdots\!95\)\( T^{9} + \)\(24\!\cdots\!22\)\( T^{10} + \)\(18\!\cdots\!73\)\( T^{11} + \)\(10\!\cdots\!72\)\( T^{12} + \)\(18\!\cdots\!73\)\( p^{3} T^{13} + \)\(24\!\cdots\!22\)\( p^{6} T^{14} + \)\(42\!\cdots\!95\)\( p^{9} T^{15} + \)\(51\!\cdots\!27\)\( p^{12} T^{16} + 63959311909282630772 p^{15} T^{17} + 903018661516769731 p^{18} T^{18} + 91665334499364 p^{21} T^{19} + 1246496536554 p^{24} T^{20} + 66383959 p^{27} T^{21} + 1433195 p^{30} T^{22} - 195 p^{33} T^{23} + p^{36} T^{24} \) | |
| 79 | \( 1 - 1299 T + 5275660 T^{2} - 5584484106 T^{3} + 12619495555432 T^{4} - 11309097156750806 T^{5} + 18521977739258441816 T^{6} - \)\(14\!\cdots\!97\)\( T^{7} + \)\(18\!\cdots\!39\)\( T^{8} - \)\(12\!\cdots\!26\)\( T^{9} + \)\(14\!\cdots\!00\)\( T^{10} - \)\(83\!\cdots\!76\)\( T^{11} + \)\(79\!\cdots\!36\)\( T^{12} - \)\(83\!\cdots\!76\)\( p^{3} T^{13} + \)\(14\!\cdots\!00\)\( p^{6} T^{14} - \)\(12\!\cdots\!26\)\( p^{9} T^{15} + \)\(18\!\cdots\!39\)\( p^{12} T^{16} - \)\(14\!\cdots\!97\)\( p^{15} T^{17} + 18521977739258441816 p^{18} T^{18} - 11309097156750806 p^{21} T^{19} + 12619495555432 p^{24} T^{20} - 5584484106 p^{27} T^{21} + 5275660 p^{30} T^{22} - 1299 p^{33} T^{23} + p^{36} T^{24} \) | |
| 83 | \( 1 - 3533 T + 9545506 T^{2} - 18633327006 T^{3} + 31311773375422 T^{4} - 44684000460681466 T^{5} + 57360084994402110298 T^{6} - \)\(65\!\cdots\!47\)\( T^{7} + \)\(69\!\cdots\!23\)\( T^{8} - \)\(67\!\cdots\!42\)\( T^{9} + \)\(60\!\cdots\!88\)\( T^{10} - \)\(50\!\cdots\!88\)\( T^{11} + \)\(39\!\cdots\!44\)\( T^{12} - \)\(50\!\cdots\!88\)\( p^{3} T^{13} + \)\(60\!\cdots\!88\)\( p^{6} T^{14} - \)\(67\!\cdots\!42\)\( p^{9} T^{15} + \)\(69\!\cdots\!23\)\( p^{12} T^{16} - \)\(65\!\cdots\!47\)\( p^{15} T^{17} + 57360084994402110298 p^{18} T^{18} - 44684000460681466 p^{21} T^{19} + 31311773375422 p^{24} T^{20} - 18633327006 p^{27} T^{21} + 9545506 p^{30} T^{22} - 3533 p^{33} T^{23} + p^{36} T^{24} \) | |
| 89 | \( 1 - 19 p T + 5236644 T^{2} - 6509895912 T^{3} + 12713212446262 T^{4} - 13207652080290262 T^{5} + 20413026778980746920 T^{6} - \)\(18\!\cdots\!87\)\( T^{7} + \)\(24\!\cdots\!08\)\( T^{8} - \)\(20\!\cdots\!57\)\( T^{9} + \)\(23\!\cdots\!08\)\( T^{10} - \)\(17\!\cdots\!18\)\( T^{11} + \)\(18\!\cdots\!28\)\( T^{12} - \)\(17\!\cdots\!18\)\( p^{3} T^{13} + \)\(23\!\cdots\!08\)\( p^{6} T^{14} - \)\(20\!\cdots\!57\)\( p^{9} T^{15} + \)\(24\!\cdots\!08\)\( p^{12} T^{16} - \)\(18\!\cdots\!87\)\( p^{15} T^{17} + 20413026778980746920 p^{18} T^{18} - 13207652080290262 p^{21} T^{19} + 12713212446262 p^{24} T^{20} - 6509895912 p^{27} T^{21} + 5236644 p^{30} T^{22} - 19 p^{34} T^{23} + p^{36} T^{24} \) | |
| 97 | \( 1 + 1178 T + 4836961 T^{2} + 3377299268 T^{3} + 11107669061723 T^{4} + 4998444616236482 T^{5} + 17794911471727511240 T^{6} + \)\(41\!\cdots\!66\)\( T^{7} + \)\(22\!\cdots\!69\)\( T^{8} + \)\(21\!\cdots\!92\)\( T^{9} + \)\(24\!\cdots\!59\)\( T^{10} + \)\(70\!\cdots\!66\)\( T^{11} + \)\(23\!\cdots\!06\)\( T^{12} + \)\(70\!\cdots\!66\)\( p^{3} T^{13} + \)\(24\!\cdots\!59\)\( p^{6} T^{14} + \)\(21\!\cdots\!92\)\( p^{9} T^{15} + \)\(22\!\cdots\!69\)\( p^{12} T^{16} + \)\(41\!\cdots\!66\)\( p^{15} T^{17} + 17794911471727511240 p^{18} T^{18} + 4998444616236482 p^{21} T^{19} + 11107669061723 p^{24} T^{20} + 3377299268 p^{27} T^{21} + 4836961 p^{30} T^{22} + 1178 p^{33} T^{23} + p^{36} T^{24} \) | |
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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
−3.89608484943097835837976768728, −3.75868004461053096390859122611, −3.53863013710723137711248444139, −3.46237466762965521529113359903, −3.41005234244663355101645666477, −3.29824472145474654961776731130, −3.26506830830876439677417243692, −3.09026992264759315047658407045, −3.08510144924250927352668647002, −2.90265645158335248289986230982, −2.89513804228909665835464414571, −2.28227831406411766353241003540, −2.21810840028662351955319489505, −2.13930041511961945948194888051, −2.09325479136097538136773423780, −2.01364419267084427380538885097, −1.61009860467738825999121305501, −1.53429497970313206893101150711, −1.52596983640189390320285304008, −1.16068463388644650858757985079, −1.02010011340707515973170641464, −0.928757833063553137085463633016, −0.70991968917996623273026441356, −0.54327971827016387231099799935, −0.54208553515682206482130587173, 0.54208553515682206482130587173, 0.54327971827016387231099799935, 0.70991968917996623273026441356, 0.928757833063553137085463633016, 1.02010011340707515973170641464, 1.16068463388644650858757985079, 1.52596983640189390320285304008, 1.53429497970313206893101150711, 1.61009860467738825999121305501, 2.01364419267084427380538885097, 2.09325479136097538136773423780, 2.13930041511961945948194888051, 2.21810840028662351955319489505, 2.28227831406411766353241003540, 2.89513804228909665835464414571, 2.90265645158335248289986230982, 3.08510144924250927352668647002, 3.09026992264759315047658407045, 3.26506830830876439677417243692, 3.29824472145474654961776731130, 3.41005234244663355101645666477, 3.46237466762965521529113359903, 3.53863013710723137711248444139, 3.75868004461053096390859122611, 3.89608484943097835837976768728
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.