Dirichlet series
| L(s) = 1 | − 48·2-s + 12·3-s + 960·4-s + 124·5-s − 576·6-s − 1.03e3·7-s − 7.16e3·8-s + 3.01e3·9-s − 5.95e3·10-s − 1.17e4·11-s + 1.15e4·12-s + 1.73e4·13-s + 4.97e4·14-s + 1.48e3·15-s − 8.60e4·16-s + 4.52e4·17-s − 1.44e5·18-s − 2.62e4·19-s + 1.19e5·20-s − 1.24e4·21-s + 5.64e5·22-s + 7.65e4·23-s − 8.60e4·24-s + 1.25e5·25-s − 8.34e5·26-s + 2.35e5·27-s − 9.94e5·28-s + ⋯ |
| L(s) = 1 | − 4.24·2-s + 0.256·3-s + 15/2·4-s + 0.443·5-s − 1.08·6-s − 1.14·7-s − 4.94·8-s + 1.37·9-s − 1.88·10-s − 2.66·11-s + 1.92·12-s + 2.19·13-s + 4.84·14-s + 0.113·15-s − 5.25·16-s + 2.23·17-s − 5.85·18-s − 0.878·19-s + 3.32·20-s − 0.292·21-s + 11.3·22-s + 1.31·23-s − 1.27·24-s + 1.60·25-s − 9.31·26-s + 2.30·27-s − 8.56·28-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 19^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(8-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 19^{12}\right)^{s/2} \, \Gamma_{\C}(s+7/2)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(24\) |
| Conductor: | \(2^{12} \cdot 19^{12}\) |
| Sign: | $1$ |
| Analytic conductor: | \(7.82862\times 10^{12}\) |
| Root analytic conductor: | \(3.44537\) |
| Motivic weight: | \(7\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((24,\ 2^{12} \cdot 19^{12} ,\ ( \ : [7/2]^{12} ),\ 1 )\) |
Particular Values
| \(L(4)\) | \(\approx\) | \(1.450884198\) |
| \(L(\frac12)\) | \(\approx\) | \(1.450884198\) |
| \(L(\frac{9}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 2 | \( ( 1 + p^{3} T + p^{6} T^{2} )^{6} \) |
| 19 | \( 1 + 26264 T + 2570714186 T^{2} + 3404714148600 p T^{3} + 387117400138786 p^{3} T^{4} + 1584226022990176 p^{6} T^{5} + 6401503235856226 p^{9} T^{6} + 1584226022990176 p^{13} T^{7} + 387117400138786 p^{17} T^{8} + 3404714148600 p^{22} T^{9} + 2570714186 p^{28} T^{10} + 26264 p^{35} T^{11} + p^{42} T^{12} \) | |
| good | 3 | \( 1 - 4 p T - 2873 T^{2} - 54860 p T^{3} + 4066231 T^{4} + 83118080 p^{2} T^{5} - 140293384 p^{2} T^{6} - 15641359616 p^{4} T^{7} - 73082822495 p^{6} T^{8} + 4133371122412 p^{6} T^{9} + 274138550251673 p^{6} T^{10} - 638929837400860 p^{8} T^{11} - 66394432008785666 p^{8} T^{12} - 638929837400860 p^{15} T^{13} + 274138550251673 p^{20} T^{14} + 4133371122412 p^{27} T^{15} - 73082822495 p^{34} T^{16} - 15641359616 p^{39} T^{17} - 140293384 p^{44} T^{18} + 83118080 p^{51} T^{19} + 4066231 p^{56} T^{20} - 54860 p^{64} T^{21} - 2873 p^{70} T^{22} - 4 p^{78} T^{23} + p^{84} T^{24} \) |
| 5 | \( 1 - 124 T - 109953 T^{2} - 426128 p^{2} T^{3} + 13660736644 T^{4} + 2491824506468 T^{5} - 951717995179701 T^{6} - 69681042997969872 p T^{7} + 681840009214422382 p^{2} T^{8} + \)\(20\!\cdots\!48\)\( p^{3} T^{9} + \)\(76\!\cdots\!19\)\( p^{4} T^{10} - \)\(79\!\cdots\!48\)\( p^{6} T^{11} - \)\(91\!\cdots\!64\)\( p^{8} T^{12} - \)\(79\!\cdots\!48\)\( p^{13} T^{13} + \)\(76\!\cdots\!19\)\( p^{18} T^{14} + \)\(20\!\cdots\!48\)\( p^{24} T^{15} + 681840009214422382 p^{30} T^{16} - 69681042997969872 p^{36} T^{17} - 951717995179701 p^{42} T^{18} + 2491824506468 p^{49} T^{19} + 13660736644 p^{56} T^{20} - 426128 p^{65} T^{21} - 109953 p^{70} T^{22} - 124 p^{77} T^{23} + p^{84} T^{24} \) | |
| 7 | \( ( 1 + 74 p T + 2580708 T^{2} + 83662286 p T^{3} + 3650249323703 T^{4} + 106363515255860 p T^{5} + 3757269319629230376 T^{6} + 106363515255860 p^{8} T^{7} + 3650249323703 p^{14} T^{8} + 83662286 p^{22} T^{9} + 2580708 p^{28} T^{10} + 74 p^{36} T^{11} + p^{42} T^{12} )^{2} \) | |
| 11 | \( ( 1 + 5880 T + 63552354 T^{2} + 235904599808 T^{3} + 1242892621867374 T^{4} + 2953394425115410992 T^{5} + \)\(15\!\cdots\!62\)\( T^{6} + 2953394425115410992 p^{7} T^{7} + 1242892621867374 p^{14} T^{8} + 235904599808 p^{21} T^{9} + 63552354 p^{28} T^{10} + 5880 p^{35} T^{11} + p^{42} T^{12} )^{2} \) | |
| 13 | \( 1 - 17390 T - 24450759 T^{2} - 147352515886 T^{3} + 30076752641644770 T^{4} - 55347895095666983378 T^{5} - \)\(11\!\cdots\!65\)\( T^{6} - \)\(19\!\cdots\!22\)\( T^{7} + \)\(11\!\cdots\!12\)\( T^{8} + \)\(73\!\cdots\!42\)\( T^{9} + \)\(87\!\cdots\!73\)\( T^{10} - \)\(76\!\cdots\!22\)\( T^{11} - \)\(24\!\cdots\!48\)\( T^{12} - \)\(76\!\cdots\!22\)\( p^{7} T^{13} + \)\(87\!\cdots\!73\)\( p^{14} T^{14} + \)\(73\!\cdots\!42\)\( p^{21} T^{15} + \)\(11\!\cdots\!12\)\( p^{28} T^{16} - \)\(19\!\cdots\!22\)\( p^{35} T^{17} - \)\(11\!\cdots\!65\)\( p^{42} T^{18} - 55347895095666983378 p^{49} T^{19} + 30076752641644770 p^{56} T^{20} - 147352515886 p^{63} T^{21} - 24450759 p^{70} T^{22} - 17390 p^{77} T^{23} + p^{84} T^{24} \) | |
| 17 | \( 1 - 2662 p T - 234580727 T^{2} + 38700949741762 T^{3} - 254353422176226298 T^{4} - \)\(10\!\cdots\!22\)\( T^{5} - \)\(52\!\cdots\!89\)\( T^{6} + \)\(15\!\cdots\!02\)\( p T^{7} + \)\(17\!\cdots\!92\)\( T^{8} - \)\(27\!\cdots\!38\)\( T^{9} - \)\(62\!\cdots\!79\)\( T^{10} + \)\(80\!\cdots\!26\)\( T^{11} + \)\(12\!\cdots\!00\)\( T^{12} + \)\(80\!\cdots\!26\)\( p^{7} T^{13} - \)\(62\!\cdots\!79\)\( p^{14} T^{14} - \)\(27\!\cdots\!38\)\( p^{21} T^{15} + \)\(17\!\cdots\!92\)\( p^{28} T^{16} + \)\(15\!\cdots\!02\)\( p^{36} T^{17} - \)\(52\!\cdots\!89\)\( p^{42} T^{18} - \)\(10\!\cdots\!22\)\( p^{49} T^{19} - 254353422176226298 p^{56} T^{20} + 38700949741762 p^{63} T^{21} - 234580727 p^{70} T^{22} - 2662 p^{78} T^{23} + p^{84} T^{24} \) | |
| 23 | \( 1 - 76554 T - 8951711043 T^{2} - 29916559914278 T^{3} + \)\(10\!\cdots\!96\)\( T^{4} + \)\(16\!\cdots\!42\)\( T^{5} - \)\(37\!\cdots\!47\)\( T^{6} - \)\(33\!\cdots\!30\)\( T^{7} + \)\(94\!\cdots\!50\)\( T^{8} + \)\(11\!\cdots\!58\)\( T^{9} + \)\(50\!\cdots\!57\)\( T^{10} - \)\(29\!\cdots\!42\)\( T^{11} - \)\(20\!\cdots\!96\)\( T^{12} - \)\(29\!\cdots\!42\)\( p^{7} T^{13} + \)\(50\!\cdots\!57\)\( p^{14} T^{14} + \)\(11\!\cdots\!58\)\( p^{21} T^{15} + \)\(94\!\cdots\!50\)\( p^{28} T^{16} - \)\(33\!\cdots\!30\)\( p^{35} T^{17} - \)\(37\!\cdots\!47\)\( p^{42} T^{18} + \)\(16\!\cdots\!42\)\( p^{49} T^{19} + \)\(10\!\cdots\!96\)\( p^{56} T^{20} - 29916559914278 p^{63} T^{21} - 8951711043 p^{70} T^{22} - 76554 p^{77} T^{23} + p^{84} T^{24} \) | |
| 29 | \( 1 - 109808 T - 35456391561 T^{2} + 1610106457469900 T^{3} + \)\(72\!\cdots\!04\)\( T^{4} + \)\(33\!\cdots\!76\)\( T^{5} - \)\(89\!\cdots\!33\)\( T^{6} - \)\(16\!\cdots\!28\)\( T^{7} + \)\(49\!\cdots\!30\)\( T^{8} + \)\(20\!\cdots\!32\)\( T^{9} + \)\(28\!\cdots\!19\)\( T^{10} - \)\(10\!\cdots\!72\)\( T^{11} - \)\(86\!\cdots\!04\)\( T^{12} - \)\(10\!\cdots\!72\)\( p^{7} T^{13} + \)\(28\!\cdots\!19\)\( p^{14} T^{14} + \)\(20\!\cdots\!32\)\( p^{21} T^{15} + \)\(49\!\cdots\!30\)\( p^{28} T^{16} - \)\(16\!\cdots\!28\)\( p^{35} T^{17} - \)\(89\!\cdots\!33\)\( p^{42} T^{18} + \)\(33\!\cdots\!76\)\( p^{49} T^{19} + \)\(72\!\cdots\!04\)\( p^{56} T^{20} + 1610106457469900 p^{63} T^{21} - 35456391561 p^{70} T^{22} - 109808 p^{77} T^{23} + p^{84} T^{24} \) | |
| 31 | \( ( 1 + 183502 T + 134104602712 T^{2} + 18526572014620050 T^{3} + \)\(79\!\cdots\!15\)\( T^{4} + \)\(87\!\cdots\!08\)\( T^{5} + \)\(27\!\cdots\!04\)\( T^{6} + \)\(87\!\cdots\!08\)\( p^{7} T^{7} + \)\(79\!\cdots\!15\)\( p^{14} T^{8} + 18526572014620050 p^{21} T^{9} + 134104602712 p^{28} T^{10} + 183502 p^{35} T^{11} + p^{42} T^{12} )^{2} \) | |
| 37 | \( ( 1 - 461070 T + 547569013512 T^{2} - 185773420236593742 T^{3} + \)\(12\!\cdots\!07\)\( T^{4} - \)\(32\!\cdots\!92\)\( T^{5} + \)\(15\!\cdots\!20\)\( T^{6} - \)\(32\!\cdots\!92\)\( p^{7} T^{7} + \)\(12\!\cdots\!07\)\( p^{14} T^{8} - 185773420236593742 p^{21} T^{9} + 547569013512 p^{28} T^{10} - 461070 p^{35} T^{11} + p^{42} T^{12} )^{2} \) | |
| 41 | \( 1 - 489142 T + 35305801619 T^{2} - 54701641104991538 T^{3} + \)\(37\!\cdots\!19\)\( T^{4} - \)\(34\!\cdots\!28\)\( T^{5} + \)\(95\!\cdots\!60\)\( T^{6} - \)\(24\!\cdots\!40\)\( T^{7} - \)\(13\!\cdots\!51\)\( T^{8} + \)\(79\!\cdots\!50\)\( T^{9} - \)\(37\!\cdots\!99\)\( T^{10} + \)\(15\!\cdots\!62\)\( T^{11} - \)\(18\!\cdots\!46\)\( T^{12} + \)\(15\!\cdots\!62\)\( p^{7} T^{13} - \)\(37\!\cdots\!99\)\( p^{14} T^{14} + \)\(79\!\cdots\!50\)\( p^{21} T^{15} - \)\(13\!\cdots\!51\)\( p^{28} T^{16} - \)\(24\!\cdots\!40\)\( p^{35} T^{17} + \)\(95\!\cdots\!60\)\( p^{42} T^{18} - \)\(34\!\cdots\!28\)\( p^{49} T^{19} + \)\(37\!\cdots\!19\)\( p^{56} T^{20} - 54701641104991538 p^{63} T^{21} + 35305801619 p^{70} T^{22} - 489142 p^{77} T^{23} + p^{84} T^{24} \) | |
| 43 | \( 1 - 1781092 T + 981909160659 T^{2} + 58766133031559564 T^{3} - \)\(27\!\cdots\!74\)\( T^{4} + \)\(17\!\cdots\!64\)\( T^{5} - \)\(10\!\cdots\!07\)\( T^{6} + \)\(25\!\cdots\!32\)\( T^{7} + \)\(31\!\cdots\!36\)\( T^{8} - \)\(21\!\cdots\!96\)\( T^{9} + \)\(11\!\cdots\!95\)\( T^{10} + \)\(18\!\cdots\!88\)\( T^{11} - \)\(55\!\cdots\!24\)\( T^{12} + \)\(18\!\cdots\!88\)\( p^{7} T^{13} + \)\(11\!\cdots\!95\)\( p^{14} T^{14} - \)\(21\!\cdots\!96\)\( p^{21} T^{15} + \)\(31\!\cdots\!36\)\( p^{28} T^{16} + \)\(25\!\cdots\!32\)\( p^{35} T^{17} - \)\(10\!\cdots\!07\)\( p^{42} T^{18} + \)\(17\!\cdots\!64\)\( p^{49} T^{19} - \)\(27\!\cdots\!74\)\( p^{56} T^{20} + 58766133031559564 p^{63} T^{21} + 981909160659 p^{70} T^{22} - 1781092 p^{77} T^{23} + p^{84} T^{24} \) | |
| 47 | \( 1 - 178950 T - 1751674983303 T^{2} + 955597726565802046 T^{3} + \)\(16\!\cdots\!96\)\( T^{4} - \)\(13\!\cdots\!86\)\( T^{5} - \)\(68\!\cdots\!99\)\( T^{6} + \)\(25\!\cdots\!06\)\( p T^{7} - \)\(62\!\cdots\!98\)\( T^{8} - \)\(62\!\cdots\!94\)\( T^{9} + \)\(33\!\cdots\!21\)\( T^{10} + \)\(14\!\cdots\!66\)\( T^{11} - \)\(23\!\cdots\!24\)\( T^{12} + \)\(14\!\cdots\!66\)\( p^{7} T^{13} + \)\(33\!\cdots\!21\)\( p^{14} T^{14} - \)\(62\!\cdots\!94\)\( p^{21} T^{15} - \)\(62\!\cdots\!98\)\( p^{28} T^{16} + \)\(25\!\cdots\!06\)\( p^{36} T^{17} - \)\(68\!\cdots\!99\)\( p^{42} T^{18} - \)\(13\!\cdots\!86\)\( p^{49} T^{19} + \)\(16\!\cdots\!96\)\( p^{56} T^{20} + 955597726565802046 p^{63} T^{21} - 1751674983303 p^{70} T^{22} - 178950 p^{77} T^{23} + p^{84} T^{24} \) | |
| 53 | \( 1 - 17734 p T - 5305961360983 T^{2} + 3598919025982818242 T^{3} + \)\(17\!\cdots\!94\)\( T^{4} - \)\(79\!\cdots\!54\)\( T^{5} - \)\(40\!\cdots\!05\)\( T^{6} + \)\(12\!\cdots\!54\)\( T^{7} + \)\(73\!\cdots\!08\)\( T^{8} - \)\(12\!\cdots\!10\)\( T^{9} - \)\(11\!\cdots\!23\)\( T^{10} + \)\(59\!\cdots\!18\)\( T^{11} + \)\(14\!\cdots\!20\)\( T^{12} + \)\(59\!\cdots\!18\)\( p^{7} T^{13} - \)\(11\!\cdots\!23\)\( p^{14} T^{14} - \)\(12\!\cdots\!10\)\( p^{21} T^{15} + \)\(73\!\cdots\!08\)\( p^{28} T^{16} + \)\(12\!\cdots\!54\)\( p^{35} T^{17} - \)\(40\!\cdots\!05\)\( p^{42} T^{18} - \)\(79\!\cdots\!54\)\( p^{49} T^{19} + \)\(17\!\cdots\!94\)\( p^{56} T^{20} + 3598919025982818242 p^{63} T^{21} - 5305961360983 p^{70} T^{22} - 17734 p^{78} T^{23} + p^{84} T^{24} \) | |
| 59 | \( 1 - 1874564 T - 1173604027217 T^{2} + 2408576674749842068 T^{3} - \)\(47\!\cdots\!81\)\( T^{4} + \)\(17\!\cdots\!04\)\( T^{5} + \)\(15\!\cdots\!48\)\( T^{6} - \)\(45\!\cdots\!72\)\( T^{7} + \)\(43\!\cdots\!49\)\( T^{8} - \)\(58\!\cdots\!12\)\( T^{9} + \)\(19\!\cdots\!93\)\( T^{10} + \)\(24\!\cdots\!92\)\( T^{11} - \)\(56\!\cdots\!54\)\( T^{12} + \)\(24\!\cdots\!92\)\( p^{7} T^{13} + \)\(19\!\cdots\!93\)\( p^{14} T^{14} - \)\(58\!\cdots\!12\)\( p^{21} T^{15} + \)\(43\!\cdots\!49\)\( p^{28} T^{16} - \)\(45\!\cdots\!72\)\( p^{35} T^{17} + \)\(15\!\cdots\!48\)\( p^{42} T^{18} + \)\(17\!\cdots\!04\)\( p^{49} T^{19} - \)\(47\!\cdots\!81\)\( p^{56} T^{20} + 2408576674749842068 p^{63} T^{21} - 1173604027217 p^{70} T^{22} - 1874564 p^{77} T^{23} + p^{84} T^{24} \) | |
| 61 | \( 1 + 1791828 T - 12651490872897 T^{2} - 16844355957694144496 T^{3} + \)\(10\!\cdots\!12\)\( T^{4} + \)\(92\!\cdots\!92\)\( T^{5} - \)\(57\!\cdots\!25\)\( T^{6} - \)\(32\!\cdots\!20\)\( T^{7} + \)\(25\!\cdots\!34\)\( T^{8} + \)\(74\!\cdots\!76\)\( T^{9} - \)\(95\!\cdots\!85\)\( T^{10} - \)\(90\!\cdots\!76\)\( T^{11} + \)\(31\!\cdots\!48\)\( T^{12} - \)\(90\!\cdots\!76\)\( p^{7} T^{13} - \)\(95\!\cdots\!85\)\( p^{14} T^{14} + \)\(74\!\cdots\!76\)\( p^{21} T^{15} + \)\(25\!\cdots\!34\)\( p^{28} T^{16} - \)\(32\!\cdots\!20\)\( p^{35} T^{17} - \)\(57\!\cdots\!25\)\( p^{42} T^{18} + \)\(92\!\cdots\!92\)\( p^{49} T^{19} + \)\(10\!\cdots\!12\)\( p^{56} T^{20} - 16844355957694144496 p^{63} T^{21} - 12651490872897 p^{70} T^{22} + 1791828 p^{77} T^{23} + p^{84} T^{24} \) | |
| 67 | \( 1 - 5363976 T - 5204027168013 T^{2} + 30379245409182544184 T^{3} + \)\(17\!\cdots\!27\)\( T^{4} - \)\(22\!\cdots\!52\)\( T^{5} - \)\(17\!\cdots\!56\)\( T^{6} + \)\(18\!\cdots\!48\)\( T^{7} + \)\(10\!\cdots\!49\)\( T^{8} + \)\(56\!\cdots\!80\)\( T^{9} - \)\(47\!\cdots\!83\)\( T^{10} - \)\(38\!\cdots\!44\)\( T^{11} + \)\(29\!\cdots\!34\)\( T^{12} - \)\(38\!\cdots\!44\)\( p^{7} T^{13} - \)\(47\!\cdots\!83\)\( p^{14} T^{14} + \)\(56\!\cdots\!80\)\( p^{21} T^{15} + \)\(10\!\cdots\!49\)\( p^{28} T^{16} + \)\(18\!\cdots\!48\)\( p^{35} T^{17} - \)\(17\!\cdots\!56\)\( p^{42} T^{18} - \)\(22\!\cdots\!52\)\( p^{49} T^{19} + \)\(17\!\cdots\!27\)\( p^{56} T^{20} + 30379245409182544184 p^{63} T^{21} - 5204027168013 p^{70} T^{22} - 5363976 p^{77} T^{23} + p^{84} T^{24} \) | |
| 71 | \( 1 - 3980204 T - 28730106281593 T^{2} + 18053900623633878380 T^{3} + \)\(90\!\cdots\!22\)\( T^{4} + \)\(33\!\cdots\!52\)\( T^{5} - \)\(11\!\cdots\!99\)\( T^{6} - \)\(26\!\cdots\!28\)\( T^{7} + \)\(10\!\cdots\!16\)\( T^{8} + \)\(29\!\cdots\!40\)\( T^{9} - \)\(18\!\cdots\!65\)\( T^{10} - \)\(16\!\cdots\!40\)\( T^{11} - \)\(42\!\cdots\!84\)\( T^{12} - \)\(16\!\cdots\!40\)\( p^{7} T^{13} - \)\(18\!\cdots\!65\)\( p^{14} T^{14} + \)\(29\!\cdots\!40\)\( p^{21} T^{15} + \)\(10\!\cdots\!16\)\( p^{28} T^{16} - \)\(26\!\cdots\!28\)\( p^{35} T^{17} - \)\(11\!\cdots\!99\)\( p^{42} T^{18} + \)\(33\!\cdots\!52\)\( p^{49} T^{19} + \)\(90\!\cdots\!22\)\( p^{56} T^{20} + 18053900623633878380 p^{63} T^{21} - 28730106281593 p^{70} T^{22} - 3980204 p^{77} T^{23} + p^{84} T^{24} \) | |
| 73 | \( 1 - 15279830 T + 81389604693739 T^{2} - \)\(19\!\cdots\!18\)\( T^{3} + \)\(99\!\cdots\!03\)\( T^{4} - \)\(85\!\cdots\!92\)\( T^{5} + \)\(33\!\cdots\!88\)\( T^{6} - \)\(76\!\cdots\!28\)\( T^{7} + \)\(34\!\cdots\!61\)\( T^{8} - \)\(17\!\cdots\!14\)\( T^{9} + \)\(57\!\cdots\!37\)\( T^{10} - \)\(17\!\cdots\!38\)\( T^{11} + \)\(59\!\cdots\!50\)\( T^{12} - \)\(17\!\cdots\!38\)\( p^{7} T^{13} + \)\(57\!\cdots\!37\)\( p^{14} T^{14} - \)\(17\!\cdots\!14\)\( p^{21} T^{15} + \)\(34\!\cdots\!61\)\( p^{28} T^{16} - \)\(76\!\cdots\!28\)\( p^{35} T^{17} + \)\(33\!\cdots\!88\)\( p^{42} T^{18} - \)\(85\!\cdots\!92\)\( p^{49} T^{19} + \)\(99\!\cdots\!03\)\( p^{56} T^{20} - \)\(19\!\cdots\!18\)\( p^{63} T^{21} + 81389604693739 p^{70} T^{22} - 15279830 p^{77} T^{23} + p^{84} T^{24} \) | |
| 79 | \( 1 + 7164236 T - 27621575053969 T^{2} - \)\(48\!\cdots\!92\)\( T^{3} - \)\(86\!\cdots\!38\)\( T^{4} + \)\(11\!\cdots\!08\)\( T^{5} + \)\(59\!\cdots\!37\)\( T^{6} - \)\(15\!\cdots\!96\)\( T^{7} - \)\(10\!\cdots\!64\)\( T^{8} - \)\(39\!\cdots\!40\)\( T^{9} - \)\(28\!\cdots\!21\)\( T^{10} + \)\(49\!\cdots\!92\)\( T^{11} + \)\(33\!\cdots\!32\)\( T^{12} + \)\(49\!\cdots\!92\)\( p^{7} T^{13} - \)\(28\!\cdots\!21\)\( p^{14} T^{14} - \)\(39\!\cdots\!40\)\( p^{21} T^{15} - \)\(10\!\cdots\!64\)\( p^{28} T^{16} - \)\(15\!\cdots\!96\)\( p^{35} T^{17} + \)\(59\!\cdots\!37\)\( p^{42} T^{18} + \)\(11\!\cdots\!08\)\( p^{49} T^{19} - \)\(86\!\cdots\!38\)\( p^{56} T^{20} - \)\(48\!\cdots\!92\)\( p^{63} T^{21} - 27621575053969 p^{70} T^{22} + 7164236 p^{77} T^{23} + p^{84} T^{24} \) | |
| 83 | \( ( 1 + 6458404 T + 78509530003758 T^{2} + \)\(41\!\cdots\!04\)\( T^{3} + \)\(33\!\cdots\!78\)\( T^{4} + \)\(13\!\cdots\!32\)\( T^{5} + \)\(98\!\cdots\!98\)\( T^{6} + \)\(13\!\cdots\!32\)\( p^{7} T^{7} + \)\(33\!\cdots\!78\)\( p^{14} T^{8} + \)\(41\!\cdots\!04\)\( p^{21} T^{9} + 78509530003758 p^{28} T^{10} + 6458404 p^{35} T^{11} + p^{42} T^{12} )^{2} \) | |
| 89 | \( 1 - 55798 p T - 162235647182159 T^{2} + \)\(63\!\cdots\!98\)\( T^{3} + \)\(14\!\cdots\!38\)\( T^{4} - \)\(42\!\cdots\!38\)\( T^{5} - \)\(95\!\cdots\!37\)\( T^{6} + \)\(25\!\cdots\!66\)\( T^{7} + \)\(48\!\cdots\!92\)\( T^{8} - \)\(10\!\cdots\!58\)\( T^{9} - \)\(22\!\cdots\!47\)\( T^{10} + \)\(18\!\cdots\!06\)\( T^{11} + \)\(10\!\cdots\!20\)\( T^{12} + \)\(18\!\cdots\!06\)\( p^{7} T^{13} - \)\(22\!\cdots\!47\)\( p^{14} T^{14} - \)\(10\!\cdots\!58\)\( p^{21} T^{15} + \)\(48\!\cdots\!92\)\( p^{28} T^{16} + \)\(25\!\cdots\!66\)\( p^{35} T^{17} - \)\(95\!\cdots\!37\)\( p^{42} T^{18} - \)\(42\!\cdots\!38\)\( p^{49} T^{19} + \)\(14\!\cdots\!38\)\( p^{56} T^{20} + \)\(63\!\cdots\!98\)\( p^{63} T^{21} - 162235647182159 p^{70} T^{22} - 55798 p^{78} T^{23} + p^{84} T^{24} \) | |
| 97 | \( 1 + 10277950 T - 169949822062069 T^{2} + \)\(13\!\cdots\!38\)\( T^{3} + \)\(51\!\cdots\!35\)\( T^{4} - \)\(29\!\cdots\!64\)\( T^{5} - \)\(20\!\cdots\!12\)\( T^{6} + \)\(85\!\cdots\!56\)\( T^{7} - \)\(41\!\cdots\!87\)\( T^{8} - \)\(59\!\cdots\!34\)\( T^{9} + \)\(70\!\cdots\!33\)\( T^{10} + \)\(34\!\cdots\!06\)\( T^{11} - \)\(60\!\cdots\!26\)\( T^{12} + \)\(34\!\cdots\!06\)\( p^{7} T^{13} + \)\(70\!\cdots\!33\)\( p^{14} T^{14} - \)\(59\!\cdots\!34\)\( p^{21} T^{15} - \)\(41\!\cdots\!87\)\( p^{28} T^{16} + \)\(85\!\cdots\!56\)\( p^{35} T^{17} - \)\(20\!\cdots\!12\)\( p^{42} T^{18} - \)\(29\!\cdots\!64\)\( p^{49} T^{19} + \)\(51\!\cdots\!35\)\( p^{56} T^{20} + \)\(13\!\cdots\!38\)\( p^{63} T^{21} - 169949822062069 p^{70} T^{22} + 10277950 p^{77} T^{23} + p^{84} T^{24} \) | |
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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
−4.54628315661149597133885277888, −4.32035879997399079194903721594, −4.08543523403469281443084391887, −3.88660520647079161709413061305, −3.85625829235374975024061116198, −3.78395349188598448097858573187, −3.38109322399997939734763761078, −3.19691728509211993109018991950, −3.18275310877216646872138879639, −2.78123844054864875865859779031, −2.69219503831087371435626878721, −2.59300211696128584354588851798, −2.48235980916752742725223466663, −2.18943623851267097219908406280, −1.92421238771489146140417800937, −1.54769563092593954333122062710, −1.54270326184502811225380986547, −1.26015342867424644584118144123, −1.26011374193958150886702058712, −0.894461901220283530137175177038, −0.816455887996797952012585644992, −0.73998732749255849659750472603, −0.54856515156220229765984334076, −0.38394128180282753280913173423, −0.24875947994334817248249856655, 0.24875947994334817248249856655, 0.38394128180282753280913173423, 0.54856515156220229765984334076, 0.73998732749255849659750472603, 0.816455887996797952012585644992, 0.894461901220283530137175177038, 1.26011374193958150886702058712, 1.26015342867424644584118144123, 1.54270326184502811225380986547, 1.54769563092593954333122062710, 1.92421238771489146140417800937, 2.18943623851267097219908406280, 2.48235980916752742725223466663, 2.59300211696128584354588851798, 2.69219503831087371435626878721, 2.78123844054864875865859779031, 3.18275310877216646872138879639, 3.19691728509211993109018991950, 3.38109322399997939734763761078, 3.78395349188598448097858573187, 3.85625829235374975024061116198, 3.88660520647079161709413061305, 4.08543523403469281443084391887, 4.32035879997399079194903721594, 4.54628315661149597133885277888
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.