Dirichlet series
| L(s) = 1 | + 24·2-s + 6·3-s + 312·4-s + 28·5-s + 144·6-s − 84·7-s + 2.91e3·8-s − 63·9-s + 672·10-s + 6·11-s + 1.87e3·12-s − 2.01e3·14-s + 168·15-s + 2.18e4·16-s − 56·17-s − 1.51e3·18-s + 158·19-s + 8.73e3·20-s − 504·21-s + 144·22-s + 414·23-s + 1.74e4·24-s − 119·25-s − 374·27-s − 2.62e4·28-s + 222·29-s + 4.03e3·30-s + ⋯ |
| L(s) = 1 | + 8.48·2-s + 1.15·3-s + 39·4-s + 2.50·5-s + 9.79·6-s − 4.53·7-s + 128.·8-s − 7/3·9-s + 21.2·10-s + 0.164·11-s + 45.0·12-s − 38.4·14-s + 2.89·15-s + 341.·16-s − 0.798·17-s − 19.7·18-s + 1.90·19-s + 97.6·20-s − 5.23·21-s + 1.39·22-s + 3.75·23-s + 148.·24-s − 0.951·25-s − 2.66·27-s − 176.·28-s + 1.42·29-s + 24.5·30-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 7^{12} \cdot 13^{24}\right)^{s/2} \, \Gamma_{\C}(s)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 7^{12} \cdot 13^{24}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(24\) |
| Conductor: | \(2^{12} \cdot 7^{12} \cdot 13^{24}\) |
| Sign: | $1$ |
| Analytic conductor: | \(5.47734\times 10^{25}\) |
| Root analytic conductor: | \(11.8151\) |
| Motivic weight: | \(3\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((24,\ 2^{12} \cdot 7^{12} \cdot 13^{24} ,\ ( \ : [3/2]^{12} ),\ 1 )\) |
Particular Values
| \(L(2)\) | \(\approx\) | \(8675.634361\) |
| \(L(\frac12)\) | \(\approx\) | \(8675.634361\) |
| \(L(\frac{5}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 2 | \( ( 1 - p T )^{12} \) |
| 7 | \( ( 1 + p T )^{12} \) | |
| 13 | \( 1 \) | |
| good | 3 | \( 1 - 2 p T + 11 p^{2} T^{2} - 598 T^{3} + 5636 T^{4} - 34274 T^{5} + 28516 p^{2} T^{6} - 1484590 T^{7} + 9712376 T^{8} - 17863850 p T^{9} + 104458021 p T^{10} - 1681038914 T^{11} + 8947335154 T^{12} - 1681038914 p^{3} T^{13} + 104458021 p^{7} T^{14} - 17863850 p^{10} T^{15} + 9712376 p^{12} T^{16} - 1484590 p^{15} T^{17} + 28516 p^{20} T^{18} - 34274 p^{21} T^{19} + 5636 p^{24} T^{20} - 598 p^{27} T^{21} + 11 p^{32} T^{22} - 2 p^{34} T^{23} + p^{36} T^{24} \) |
| 5 | \( 1 - 28 T + 903 T^{2} - 18336 T^{3} + 394349 T^{4} - 6456942 T^{5} + 109695033 T^{6} - 61908492 p^{2} T^{7} + 4516984072 p T^{8} - 283465723398 T^{9} + 3695305707727 T^{10} - 8453583210206 p T^{11} + 20128126868503 p^{2} T^{12} - 8453583210206 p^{4} T^{13} + 3695305707727 p^{6} T^{14} - 283465723398 p^{9} T^{15} + 4516984072 p^{13} T^{16} - 61908492 p^{17} T^{17} + 109695033 p^{18} T^{18} - 6456942 p^{21} T^{19} + 394349 p^{24} T^{20} - 18336 p^{27} T^{21} + 903 p^{30} T^{22} - 28 p^{33} T^{23} + p^{36} T^{24} \) | |
| 11 | \( 1 - 6 T + 3697 T^{2} - 133428 T^{3} + 9455751 T^{4} - 355363090 T^{5} + 23902966779 T^{6} - 725147254528 T^{7} + 38838677734375 T^{8} - 1416752415168770 T^{9} + 5255539725598852 p T^{10} - 1900023473997568206 T^{11} + 87483227287851653090 T^{12} - 1900023473997568206 p^{3} T^{13} + 5255539725598852 p^{7} T^{14} - 1416752415168770 p^{9} T^{15} + 38838677734375 p^{12} T^{16} - 725147254528 p^{15} T^{17} + 23902966779 p^{18} T^{18} - 355363090 p^{21} T^{19} + 9455751 p^{24} T^{20} - 133428 p^{27} T^{21} + 3697 p^{30} T^{22} - 6 p^{33} T^{23} + p^{36} T^{24} \) | |
| 17 | \( 1 + 56 T + 27756 T^{2} + 1830796 T^{3} + 395755848 T^{4} + 27415419396 T^{5} + 4031976122932 T^{6} + 263512597212356 T^{7} + 32210161281098744 T^{8} + 1931704097734587716 T^{9} + \)\(20\!\cdots\!20\)\( T^{10} + \)\(68\!\cdots\!96\)\( p T^{11} + \)\(11\!\cdots\!30\)\( T^{12} + \)\(68\!\cdots\!96\)\( p^{4} T^{13} + \)\(20\!\cdots\!20\)\( p^{6} T^{14} + 1931704097734587716 p^{9} T^{15} + 32210161281098744 p^{12} T^{16} + 263512597212356 p^{15} T^{17} + 4031976122932 p^{18} T^{18} + 27415419396 p^{21} T^{19} + 395755848 p^{24} T^{20} + 1830796 p^{27} T^{21} + 27756 p^{30} T^{22} + 56 p^{33} T^{23} + p^{36} T^{24} \) | |
| 19 | \( 1 - 158 T + 63961 T^{2} - 8579186 T^{3} + 1962500133 T^{4} - 228662218846 T^{5} + 38234668064881 T^{6} - 3911741655898978 T^{7} + 526477586838644543 T^{8} - 47528495663113681068 T^{9} + \)\(53\!\cdots\!66\)\( T^{10} - \)\(42\!\cdots\!32\)\( T^{11} + \)\(42\!\cdots\!02\)\( T^{12} - \)\(42\!\cdots\!32\)\( p^{3} T^{13} + \)\(53\!\cdots\!66\)\( p^{6} T^{14} - 47528495663113681068 p^{9} T^{15} + 526477586838644543 p^{12} T^{16} - 3911741655898978 p^{15} T^{17} + 38234668064881 p^{18} T^{18} - 228662218846 p^{21} T^{19} + 1962500133 p^{24} T^{20} - 8579186 p^{27} T^{21} + 63961 p^{30} T^{22} - 158 p^{33} T^{23} + p^{36} T^{24} \) | |
| 23 | \( 1 - 18 p T + 129436 T^{2} - 27872430 T^{3} + 5341169367 T^{4} - 843232151132 T^{5} + 126701110947478 T^{6} - 17001268819929452 T^{7} + 2294812606402169031 T^{8} - \)\(28\!\cdots\!38\)\( T^{9} + \)\(36\!\cdots\!78\)\( T^{10} - \)\(42\!\cdots\!42\)\( T^{11} + \)\(49\!\cdots\!46\)\( T^{12} - \)\(42\!\cdots\!42\)\( p^{3} T^{13} + \)\(36\!\cdots\!78\)\( p^{6} T^{14} - \)\(28\!\cdots\!38\)\( p^{9} T^{15} + 2294812606402169031 p^{12} T^{16} - 17001268819929452 p^{15} T^{17} + 126701110947478 p^{18} T^{18} - 843232151132 p^{21} T^{19} + 5341169367 p^{24} T^{20} - 27872430 p^{27} T^{21} + 129436 p^{30} T^{22} - 18 p^{34} T^{23} + p^{36} T^{24} \) | |
| 29 | \( 1 - 222 T + 151725 T^{2} - 20608512 T^{3} + 9539581272 T^{4} - 26406053518 p T^{5} + 385207158811726 T^{6} - 15501650562739294 T^{7} + 12450692238799250802 T^{8} - \)\(10\!\cdots\!60\)\( T^{9} + \)\(33\!\cdots\!29\)\( T^{10} + \)\(50\!\cdots\!74\)\( T^{11} + \)\(83\!\cdots\!90\)\( T^{12} + \)\(50\!\cdots\!74\)\( p^{3} T^{13} + \)\(33\!\cdots\!29\)\( p^{6} T^{14} - \)\(10\!\cdots\!60\)\( p^{9} T^{15} + 12450692238799250802 p^{12} T^{16} - 15501650562739294 p^{15} T^{17} + 385207158811726 p^{18} T^{18} - 26406053518 p^{22} T^{19} + 9539581272 p^{24} T^{20} - 20608512 p^{27} T^{21} + 151725 p^{30} T^{22} - 222 p^{33} T^{23} + p^{36} T^{24} \) | |
| 31 | \( 1 + 200 T + 211772 T^{2} + 39626874 T^{3} + 22571101195 T^{4} + 3865965334988 T^{5} + 1592531038048406 T^{6} + 247025439215952894 T^{7} + 82883164234396426887 T^{8} + \)\(11\!\cdots\!86\)\( T^{9} + \)\(33\!\cdots\!46\)\( T^{10} + \)\(42\!\cdots\!58\)\( T^{11} + \)\(11\!\cdots\!02\)\( T^{12} + \)\(42\!\cdots\!58\)\( p^{3} T^{13} + \)\(33\!\cdots\!46\)\( p^{6} T^{14} + \)\(11\!\cdots\!86\)\( p^{9} T^{15} + 82883164234396426887 p^{12} T^{16} + 247025439215952894 p^{15} T^{17} + 1592531038048406 p^{18} T^{18} + 3865965334988 p^{21} T^{19} + 22571101195 p^{24} T^{20} + 39626874 p^{27} T^{21} + 211772 p^{30} T^{22} + 200 p^{33} T^{23} + p^{36} T^{24} \) | |
| 37 | \( 1 + 560 T + 466958 T^{2} + 5209604 p T^{3} + 98589641315 T^{4} + 33834796369192 T^{5} + 13317975839077178 T^{6} + 3952519366328377292 T^{7} + \)\(12\!\cdots\!39\)\( T^{8} + \)\(91\!\cdots\!56\)\( p T^{9} + \)\(95\!\cdots\!24\)\( T^{10} + \)\(21\!\cdots\!84\)\( T^{11} + \)\(54\!\cdots\!10\)\( T^{12} + \)\(21\!\cdots\!84\)\( p^{3} T^{13} + \)\(95\!\cdots\!24\)\( p^{6} T^{14} + \)\(91\!\cdots\!56\)\( p^{10} T^{15} + \)\(12\!\cdots\!39\)\( p^{12} T^{16} + 3952519366328377292 p^{15} T^{17} + 13317975839077178 p^{18} T^{18} + 33834796369192 p^{21} T^{19} + 98589641315 p^{24} T^{20} + 5209604 p^{28} T^{21} + 466958 p^{30} T^{22} + 560 p^{33} T^{23} + p^{36} T^{24} \) | |
| 41 | \( 1 + 66 T + 309298 T^{2} + 16048924 T^{3} + 47427693159 T^{4} + 1822340795832 T^{5} + 4553488568162910 T^{6} + 113077312707467294 T^{7} + \)\(29\!\cdots\!43\)\( T^{8} + \)\(25\!\cdots\!28\)\( T^{9} + \)\(14\!\cdots\!00\)\( T^{10} - \)\(18\!\cdots\!92\)\( T^{11} + \)\(74\!\cdots\!90\)\( T^{12} - \)\(18\!\cdots\!92\)\( p^{3} T^{13} + \)\(14\!\cdots\!00\)\( p^{6} T^{14} + \)\(25\!\cdots\!28\)\( p^{9} T^{15} + \)\(29\!\cdots\!43\)\( p^{12} T^{16} + 113077312707467294 p^{15} T^{17} + 4553488568162910 p^{18} T^{18} + 1822340795832 p^{21} T^{19} + 47427693159 p^{24} T^{20} + 16048924 p^{27} T^{21} + 309298 p^{30} T^{22} + 66 p^{33} T^{23} + p^{36} T^{24} \) | |
| 43 | \( 1 - 484 T + 577521 T^{2} - 296906280 T^{3} + 177139327387 T^{4} - 85705031273194 T^{5} + 37369683128816491 T^{6} - 15740494611749708510 T^{7} + \)\(58\!\cdots\!79\)\( T^{8} - \)\(20\!\cdots\!94\)\( T^{9} + \)\(68\!\cdots\!16\)\( T^{10} - \)\(21\!\cdots\!26\)\( T^{11} + \)\(62\!\cdots\!18\)\( T^{12} - \)\(21\!\cdots\!26\)\( p^{3} T^{13} + \)\(68\!\cdots\!16\)\( p^{6} T^{14} - \)\(20\!\cdots\!94\)\( p^{9} T^{15} + \)\(58\!\cdots\!79\)\( p^{12} T^{16} - 15740494611749708510 p^{15} T^{17} + 37369683128816491 p^{18} T^{18} - 85705031273194 p^{21} T^{19} + 177139327387 p^{24} T^{20} - 296906280 p^{27} T^{21} + 577521 p^{30} T^{22} - 484 p^{33} T^{23} + p^{36} T^{24} \) | |
| 47 | \( 1 - 618 T + 1021286 T^{2} - 530903780 T^{3} + 490553614747 T^{4} - 219637031309392 T^{5} + 147616530931893554 T^{6} - 57739916506689994242 T^{7} + \)\(31\!\cdots\!07\)\( T^{8} - \)\(10\!\cdots\!16\)\( T^{9} + \)\(48\!\cdots\!24\)\( T^{10} - \)\(14\!\cdots\!36\)\( T^{11} + \)\(57\!\cdots\!98\)\( T^{12} - \)\(14\!\cdots\!36\)\( p^{3} T^{13} + \)\(48\!\cdots\!24\)\( p^{6} T^{14} - \)\(10\!\cdots\!16\)\( p^{9} T^{15} + \)\(31\!\cdots\!07\)\( p^{12} T^{16} - 57739916506689994242 p^{15} T^{17} + 147616530931893554 p^{18} T^{18} - 219637031309392 p^{21} T^{19} + 490553614747 p^{24} T^{20} - 530903780 p^{27} T^{21} + 1021286 p^{30} T^{22} - 618 p^{33} T^{23} + p^{36} T^{24} \) | |
| 53 | \( 1 - 504 T + 823364 T^{2} - 367120608 T^{3} + 339472257980 T^{4} - 139818790992912 T^{5} + 1835116751754036 p T^{6} - 36590735092084851648 T^{7} + \)\(21\!\cdots\!60\)\( T^{8} - \)\(73\!\cdots\!28\)\( T^{9} + \)\(39\!\cdots\!32\)\( T^{10} - \)\(12\!\cdots\!00\)\( T^{11} + \)\(62\!\cdots\!10\)\( T^{12} - \)\(12\!\cdots\!00\)\( p^{3} T^{13} + \)\(39\!\cdots\!32\)\( p^{6} T^{14} - \)\(73\!\cdots\!28\)\( p^{9} T^{15} + \)\(21\!\cdots\!60\)\( p^{12} T^{16} - 36590735092084851648 p^{15} T^{17} + 1835116751754036 p^{19} T^{18} - 139818790992912 p^{21} T^{19} + 339472257980 p^{24} T^{20} - 367120608 p^{27} T^{21} + 823364 p^{30} T^{22} - 504 p^{33} T^{23} + p^{36} T^{24} \) | |
| 59 | \( 1 - 1460 T + 1353892 T^{2} - 690621906 T^{3} + 203789102524 T^{4} + 33299337446536 T^{5} - 57039599631620094 T^{6} + 36304927317651161324 T^{7} - \)\(12\!\cdots\!72\)\( T^{8} + \)\(32\!\cdots\!02\)\( T^{9} + \)\(61\!\cdots\!40\)\( T^{10} - \)\(85\!\cdots\!32\)\( T^{11} + \)\(60\!\cdots\!70\)\( T^{12} - \)\(85\!\cdots\!32\)\( p^{3} T^{13} + \)\(61\!\cdots\!40\)\( p^{6} T^{14} + \)\(32\!\cdots\!02\)\( p^{9} T^{15} - \)\(12\!\cdots\!72\)\( p^{12} T^{16} + 36304927317651161324 p^{15} T^{17} - 57039599631620094 p^{18} T^{18} + 33299337446536 p^{21} T^{19} + 203789102524 p^{24} T^{20} - 690621906 p^{27} T^{21} + 1353892 p^{30} T^{22} - 1460 p^{33} T^{23} + p^{36} T^{24} \) | |
| 61 | \( 1 + 2 T + 1538090 T^{2} + 66788400 T^{3} + 1231731837592 T^{4} + 77730309979002 T^{5} + 668132476378842856 T^{6} + 47848387709629995682 T^{7} + \)\(27\!\cdots\!32\)\( T^{8} + \)\(19\!\cdots\!32\)\( T^{9} + \)\(85\!\cdots\!82\)\( T^{10} + \)\(97\!\cdots\!54\)\( p T^{11} + \)\(21\!\cdots\!10\)\( T^{12} + \)\(97\!\cdots\!54\)\( p^{4} T^{13} + \)\(85\!\cdots\!82\)\( p^{6} T^{14} + \)\(19\!\cdots\!32\)\( p^{9} T^{15} + \)\(27\!\cdots\!32\)\( p^{12} T^{16} + 47848387709629995682 p^{15} T^{17} + 668132476378842856 p^{18} T^{18} + 77730309979002 p^{21} T^{19} + 1231731837592 p^{24} T^{20} + 66788400 p^{27} T^{21} + 1538090 p^{30} T^{22} + 2 p^{33} T^{23} + p^{36} T^{24} \) | |
| 67 | \( 1 - 334 T + 588928 T^{2} - 260671602 T^{3} + 438479727663 T^{4} - 125594436453532 T^{5} + 159520194153151946 T^{6} - 46497873306441767872 T^{7} + \)\(65\!\cdots\!03\)\( T^{8} - \)\(11\!\cdots\!62\)\( T^{9} + \)\(18\!\cdots\!70\)\( T^{10} - \)\(31\!\cdots\!10\)\( T^{11} + \)\(62\!\cdots\!34\)\( T^{12} - \)\(31\!\cdots\!10\)\( p^{3} T^{13} + \)\(18\!\cdots\!70\)\( p^{6} T^{14} - \)\(11\!\cdots\!62\)\( p^{9} T^{15} + \)\(65\!\cdots\!03\)\( p^{12} T^{16} - 46497873306441767872 p^{15} T^{17} + 159520194153151946 p^{18} T^{18} - 125594436453532 p^{21} T^{19} + 438479727663 p^{24} T^{20} - 260671602 p^{27} T^{21} + 588928 p^{30} T^{22} - 334 p^{33} T^{23} + p^{36} T^{24} \) | |
| 71 | \( 1 + 196 T + 2416598 T^{2} + 394874224 T^{3} + 2921231656855 T^{4} + 433653423521304 T^{5} + 2381847254667230994 T^{6} + \)\(33\!\cdots\!96\)\( T^{7} + \)\(14\!\cdots\!31\)\( T^{8} + \)\(19\!\cdots\!00\)\( T^{9} + \)\(71\!\cdots\!04\)\( T^{10} + \)\(90\!\cdots\!84\)\( T^{11} + \)\(28\!\cdots\!14\)\( T^{12} + \)\(90\!\cdots\!84\)\( p^{3} T^{13} + \)\(71\!\cdots\!04\)\( p^{6} T^{14} + \)\(19\!\cdots\!00\)\( p^{9} T^{15} + \)\(14\!\cdots\!31\)\( p^{12} T^{16} + \)\(33\!\cdots\!96\)\( p^{15} T^{17} + 2381847254667230994 p^{18} T^{18} + 433653423521304 p^{21} T^{19} + 2921231656855 p^{24} T^{20} + 394874224 p^{27} T^{21} + 2416598 p^{30} T^{22} + 196 p^{33} T^{23} + p^{36} T^{24} \) | |
| 73 | \( 1 - 490 T + 1720898 T^{2} - 734481732 T^{3} + 1724216988631 T^{4} - 643855094776440 T^{5} + 1252260813806487742 T^{6} - \)\(39\!\cdots\!22\)\( T^{7} + \)\(71\!\cdots\!23\)\( T^{8} - \)\(19\!\cdots\!84\)\( T^{9} + \)\(34\!\cdots\!16\)\( T^{10} - \)\(85\!\cdots\!20\)\( T^{11} + \)\(14\!\cdots\!18\)\( T^{12} - \)\(85\!\cdots\!20\)\( p^{3} T^{13} + \)\(34\!\cdots\!16\)\( p^{6} T^{14} - \)\(19\!\cdots\!84\)\( p^{9} T^{15} + \)\(71\!\cdots\!23\)\( p^{12} T^{16} - \)\(39\!\cdots\!22\)\( p^{15} T^{17} + 1252260813806487742 p^{18} T^{18} - 643855094776440 p^{21} T^{19} + 1724216988631 p^{24} T^{20} - 734481732 p^{27} T^{21} + 1720898 p^{30} T^{22} - 490 p^{33} T^{23} + p^{36} T^{24} \) | |
| 79 | \( 1 - 2942 T + 7231662 T^{2} - 11873027278 T^{3} + 17403718671350 T^{4} - 20700125542468954 T^{5} + 22950613849313189206 T^{6} - \)\(22\!\cdots\!62\)\( T^{7} + \)\(20\!\cdots\!27\)\( T^{8} - \)\(17\!\cdots\!20\)\( T^{9} + \)\(14\!\cdots\!24\)\( T^{10} - \)\(10\!\cdots\!08\)\( T^{11} + \)\(78\!\cdots\!12\)\( T^{12} - \)\(10\!\cdots\!08\)\( p^{3} T^{13} + \)\(14\!\cdots\!24\)\( p^{6} T^{14} - \)\(17\!\cdots\!20\)\( p^{9} T^{15} + \)\(20\!\cdots\!27\)\( p^{12} T^{16} - \)\(22\!\cdots\!62\)\( p^{15} T^{17} + 22950613849313189206 p^{18} T^{18} - 20700125542468954 p^{21} T^{19} + 17403718671350 p^{24} T^{20} - 11873027278 p^{27} T^{21} + 7231662 p^{30} T^{22} - 2942 p^{33} T^{23} + p^{36} T^{24} \) | |
| 83 | \( 1 - 236 T + 3395476 T^{2} - 1429947482 T^{3} + 6442122063583 T^{4} - 3046498900715516 T^{5} + 8667331529809603750 T^{6} - \)\(41\!\cdots\!78\)\( T^{7} + \)\(87\!\cdots\!63\)\( T^{8} - \)\(40\!\cdots\!50\)\( T^{9} + \)\(69\!\cdots\!86\)\( T^{10} - \)\(29\!\cdots\!50\)\( T^{11} + \)\(44\!\cdots\!98\)\( T^{12} - \)\(29\!\cdots\!50\)\( p^{3} T^{13} + \)\(69\!\cdots\!86\)\( p^{6} T^{14} - \)\(40\!\cdots\!50\)\( p^{9} T^{15} + \)\(87\!\cdots\!63\)\( p^{12} T^{16} - \)\(41\!\cdots\!78\)\( p^{15} T^{17} + 8667331529809603750 p^{18} T^{18} - 3046498900715516 p^{21} T^{19} + 6442122063583 p^{24} T^{20} - 1429947482 p^{27} T^{21} + 3395476 p^{30} T^{22} - 236 p^{33} T^{23} + p^{36} T^{24} \) | |
| 89 | \( 1 - 3566 T + 12113993 T^{2} - 27150798426 T^{3} + 55870718931573 T^{4} - 93342885517388726 T^{5} + \)\(14\!\cdots\!01\)\( T^{6} - \)\(19\!\cdots\!86\)\( T^{7} + \)\(23\!\cdots\!87\)\( T^{8} - \)\(26\!\cdots\!72\)\( T^{9} + \)\(27\!\cdots\!30\)\( T^{10} - \)\(26\!\cdots\!76\)\( T^{11} + \)\(22\!\cdots\!34\)\( T^{12} - \)\(26\!\cdots\!76\)\( p^{3} T^{13} + \)\(27\!\cdots\!30\)\( p^{6} T^{14} - \)\(26\!\cdots\!72\)\( p^{9} T^{15} + \)\(23\!\cdots\!87\)\( p^{12} T^{16} - \)\(19\!\cdots\!86\)\( p^{15} T^{17} + \)\(14\!\cdots\!01\)\( p^{18} T^{18} - 93342885517388726 p^{21} T^{19} + 55870718931573 p^{24} T^{20} - 27150798426 p^{27} T^{21} + 12113993 p^{30} T^{22} - 3566 p^{33} T^{23} + p^{36} T^{24} \) | |
| 97 | \( 1 - 3032 T + 11810929 T^{2} - 22755029926 T^{3} + 49470031369093 T^{4} - 66042278457753202 T^{5} + 98865976542885127461 T^{6} - \)\(87\!\cdots\!60\)\( T^{7} + \)\(93\!\cdots\!19\)\( T^{8} - \)\(33\!\cdots\!92\)\( T^{9} + \)\(20\!\cdots\!70\)\( T^{10} + \)\(44\!\cdots\!84\)\( T^{11} - \)\(23\!\cdots\!22\)\( T^{12} + \)\(44\!\cdots\!84\)\( p^{3} T^{13} + \)\(20\!\cdots\!70\)\( p^{6} T^{14} - \)\(33\!\cdots\!92\)\( p^{9} T^{15} + \)\(93\!\cdots\!19\)\( p^{12} T^{16} - \)\(87\!\cdots\!60\)\( p^{15} T^{17} + 98865976542885127461 p^{18} T^{18} - 66042278457753202 p^{21} T^{19} + 49470031369093 p^{24} T^{20} - 22755029926 p^{27} T^{21} + 11810929 p^{30} T^{22} - 3032 p^{33} T^{23} + p^{36} T^{24} \) | |
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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
−2.55490594178410214649610257430, −2.39120675069890019927664133855, −2.31171489352338095683790263841, −2.21001466129704102201507181085, −2.14933021459430579910960882420, −2.06533869065877337775361642269, −2.02447053816468620393245344547, −2.02169436448690118046791076489, −1.99172914615187724291673790045, −1.88079027974103890082880724927, −1.81632158176703575920496528470, −1.77195904753934565508225113277, −1.46740150604001268380293177349, −1.40630812586501258734822230637, −1.11279424468566781543937864339, −1.03760229360448849598560500332, −1.03549934648400582151734169145, −0.876379579833826652199908018599, −0.74849642274820027868051563143, −0.70619601283422597875896286726, −0.63281971083060097349711633691, −0.61022969628335808658359671568, −0.50873470615502136646058164797, −0.16839695860115570035893261638, −0.06755263850285842161566565534, 0.06755263850285842161566565534, 0.16839695860115570035893261638, 0.50873470615502136646058164797, 0.61022969628335808658359671568, 0.63281971083060097349711633691, 0.70619601283422597875896286726, 0.74849642274820027868051563143, 0.876379579833826652199908018599, 1.03549934648400582151734169145, 1.03760229360448849598560500332, 1.11279424468566781543937864339, 1.40630812586501258734822230637, 1.46740150604001268380293177349, 1.77195904753934565508225113277, 1.81632158176703575920496528470, 1.88079027974103890082880724927, 1.99172914615187724291673790045, 2.02169436448690118046791076489, 2.02447053816468620393245344547, 2.06533869065877337775361642269, 2.14933021459430579910960882420, 2.21001466129704102201507181085, 2.31171489352338095683790263841, 2.39120675069890019927664133855, 2.55490594178410214649610257430
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.