| L(s) = 1 | − 4·2-s − 12·3-s + 3·4-s − 18·5-s + 48·6-s − 28·7-s + 51·9-s + 72·10-s + 44·11-s − 36·12-s − 134·13-s + 112·14-s + 216·15-s − 25·16-s − 74·17-s − 204·18-s − 164·19-s − 54·20-s + 336·21-s − 176·22-s + 194·23-s − 69·25-s + 536·26-s − 98·27-s − 84·28-s − 108·29-s − 864·30-s + ⋯ |
| L(s) = 1 | − 1.41·2-s − 2.30·3-s + 3/8·4-s − 1.60·5-s + 3.26·6-s − 1.51·7-s + 17/9·9-s + 2.27·10-s + 1.20·11-s − 0.866·12-s − 2.85·13-s + 2.13·14-s + 3.71·15-s − 0.390·16-s − 1.05·17-s − 2.67·18-s − 1.98·19-s − 0.603·20-s + 3.49·21-s − 1.70·22-s + 1.75·23-s − 0.551·25-s + 4.04·26-s − 0.698·27-s − 0.566·28-s − 0.691·29-s − 5.25·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 35153041 ^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 35153041 ^{s/2} \, \Gamma_{\C}(s+3/2)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 7 | $C_1$ | \( ( 1 + p T )^{4} \) |
| 11 | $C_1$ | \( ( 1 - p T )^{4} \) |
| good | 2 | $C_2 \wr S_4$ | \( 1 + p^{2} T + 13 T^{2} + 5 p^{3} T^{3} + 73 p T^{4} + 5 p^{6} T^{5} + 13 p^{6} T^{6} + p^{11} T^{7} + p^{12} T^{8} \) |
| 3 | $C_2 \wr S_4$ | \( 1 + 4 p T + 31 p T^{2} + 602 T^{3} + 1168 p T^{4} + 602 p^{3} T^{5} + 31 p^{7} T^{6} + 4 p^{10} T^{7} + p^{12} T^{8} \) |
| 5 | $C_2 \wr S_4$ | \( 1 + 18 T + 393 T^{2} + 758 p T^{3} + 56148 T^{4} + 758 p^{4} T^{5} + 393 p^{6} T^{6} + 18 p^{9} T^{7} + p^{12} T^{8} \) |
| 13 | $C_2 \wr S_4$ | \( 1 + 134 T + 10520 T^{2} + 559034 T^{3} + 26667070 T^{4} + 559034 p^{3} T^{5} + 10520 p^{6} T^{6} + 134 p^{9} T^{7} + p^{12} T^{8} \) |
| 17 | $C_2 \wr S_4$ | \( 1 + 74 T + 17468 T^{2} + 831750 T^{3} + 118657126 T^{4} + 831750 p^{3} T^{5} + 17468 p^{6} T^{6} + 74 p^{9} T^{7} + p^{12} T^{8} \) |
| 19 | $C_2 \wr S_4$ | \( 1 + 164 T + 11968 T^{2} + 138820 T^{3} - 15467058 T^{4} + 138820 p^{3} T^{5} + 11968 p^{6} T^{6} + 164 p^{9} T^{7} + p^{12} T^{8} \) |
| 23 | $C_2 \wr S_4$ | \( 1 - 194 T + 46053 T^{2} - 5374478 T^{3} + 784861428 T^{4} - 5374478 p^{3} T^{5} + 46053 p^{6} T^{6} - 194 p^{9} T^{7} + p^{12} T^{8} \) |
| 29 | $C_2 \wr S_4$ | \( 1 + 108 T + 42724 T^{2} - 1514908 T^{3} + 528463590 T^{4} - 1514908 p^{3} T^{5} + 42724 p^{6} T^{6} + 108 p^{9} T^{7} + p^{12} T^{8} \) |
| 31 | $C_2 \wr S_4$ | \( 1 + 412 T + 51613 T^{2} - 11046366 T^{3} - 3907770320 T^{4} - 11046366 p^{3} T^{5} + 51613 p^{6} T^{6} + 412 p^{9} T^{7} + p^{12} T^{8} \) |
| 37 | $C_2 \wr S_4$ | \( 1 - 286 T + 135513 T^{2} - 31935490 T^{3} + 9691467156 T^{4} - 31935490 p^{3} T^{5} + 135513 p^{6} T^{6} - 286 p^{9} T^{7} + p^{12} T^{8} \) |
| 41 | $C_2 \wr S_4$ | \( 1 + 18 T + 114524 T^{2} - 18203666 T^{3} + 5875037446 T^{4} - 18203666 p^{3} T^{5} + 114524 p^{6} T^{6} + 18 p^{9} T^{7} + p^{12} T^{8} \) |
| 43 | $C_2 \wr S_4$ | \( 1 + 496 T + 308268 T^{2} + 103155312 T^{3} + 35378135478 T^{4} + 103155312 p^{3} T^{5} + 308268 p^{6} T^{6} + 496 p^{9} T^{7} + p^{12} T^{8} \) |
| 47 | $C_2 \wr S_4$ | \( 1 - 62 T + 345832 T^{2} - 21522086 T^{3} + 50715680878 T^{4} - 21522086 p^{3} T^{5} + 345832 p^{6} T^{6} - 62 p^{9} T^{7} + p^{12} T^{8} \) |
| 53 | $C_2 \wr S_4$ | \( 1 + 828 T + 705892 T^{2} + 360136820 T^{3} + 165458948486 T^{4} + 360136820 p^{3} T^{5} + 705892 p^{6} T^{6} + 828 p^{9} T^{7} + p^{12} T^{8} \) |
| 59 | $C_2 \wr S_4$ | \( 1 + 1224 T + 1275413 T^{2} + 804281630 T^{3} + 441199752912 T^{4} + 804281630 p^{3} T^{5} + 1275413 p^{6} T^{6} + 1224 p^{9} T^{7} + p^{12} T^{8} \) |
| 61 | $C_2 \wr S_4$ | \( 1 + 350 T + 430588 T^{2} + 138974618 T^{3} + 88699741846 T^{4} + 138974618 p^{3} T^{5} + 430588 p^{6} T^{6} + 350 p^{9} T^{7} + p^{12} T^{8} \) |
| 67 | $C_2 \wr S_4$ | \( 1 + 1498 T + 1714917 T^{2} + 1237137334 T^{3} + 793168291468 T^{4} + 1237137334 p^{3} T^{5} + 1714917 p^{6} T^{6} + 1498 p^{9} T^{7} + p^{12} T^{8} \) |
| 71 | $C_2 \wr S_4$ | \( 1 - 2326 T + 3445789 T^{2} - 3267789594 T^{3} + 2320231641060 T^{4} - 3267789594 p^{3} T^{5} + 3445789 p^{6} T^{6} - 2326 p^{9} T^{7} + p^{12} T^{8} \) |
| 73 | $C_2 \wr S_4$ | \( 1 + 1630 T + 2286292 T^{2} + 1913055042 T^{3} + 1441611662422 T^{4} + 1913055042 p^{3} T^{5} + 2286292 p^{6} T^{6} + 1630 p^{9} T^{7} + p^{12} T^{8} \) |
| 79 | $C_2 \wr S_4$ | \( 1 + 1020 T + 1989936 T^{2} + 1295128300 T^{3} + 1427479601566 T^{4} + 1295128300 p^{3} T^{5} + 1989936 p^{6} T^{6} + 1020 p^{9} T^{7} + p^{12} T^{8} \) |
| 83 | $C_2 \wr S_4$ | \( 1 + 1920 T + 2662892 T^{2} + 2752584832 T^{3} + 2433754978902 T^{4} + 2752584832 p^{3} T^{5} + 2662892 p^{6} T^{6} + 1920 p^{9} T^{7} + p^{12} T^{8} \) |
| 89 | $C_2 \wr S_4$ | \( 1 - 1550 T + 1862281 T^{2} - 874387910 T^{3} + 705085008356 T^{4} - 874387910 p^{3} T^{5} + 1862281 p^{6} T^{6} - 1550 p^{9} T^{7} + p^{12} T^{8} \) |
| 97 | $C_2 \wr S_4$ | \( 1 + 2202 T + 5018625 T^{2} + 6116849198 T^{3} + 7416588769820 T^{4} + 6116849198 p^{3} T^{5} + 5018625 p^{6} T^{6} + 2202 p^{9} T^{7} + p^{12} T^{8} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.91650628795341302662702545800, −10.88071735527003557479835441964, −9.967921623863442184276883178611, −9.889853972950125312718133112692, −9.680371771411547379624234392329, −9.453554465884656392953467879988, −8.928873679924078421847383066045, −8.832999759798760254606689015754, −8.758606309476125421947229676998, −7.942117280545198831768836397236, −7.68448651941798088506216457941, −7.41000213023357838781104720931, −6.98274345888302426721498898799, −6.89947703437141300004680603572, −6.44833032670455203553731329261, −5.98983998150940296546754602579, −5.97240785298497489268845710408, −5.42217514794295291291569259983, −4.65625976084825097502996763721, −4.63810192447921754194094079760, −4.36507013149409486297673882603, −3.67358803512856225534933034657, −3.18550746492473304821872059558, −2.51333658246806615569385710141, −1.67315106252138622706386683089, 0, 0, 0, 0,
1.67315106252138622706386683089, 2.51333658246806615569385710141, 3.18550746492473304821872059558, 3.67358803512856225534933034657, 4.36507013149409486297673882603, 4.63810192447921754194094079760, 4.65625976084825097502996763721, 5.42217514794295291291569259983, 5.97240785298497489268845710408, 5.98983998150940296546754602579, 6.44833032670455203553731329261, 6.89947703437141300004680603572, 6.98274345888302426721498898799, 7.41000213023357838781104720931, 7.68448651941798088506216457941, 7.942117280545198831768836397236, 8.758606309476125421947229676998, 8.832999759798760254606689015754, 8.928873679924078421847383066045, 9.453554465884656392953467879988, 9.680371771411547379624234392329, 9.889853972950125312718133112692, 9.967921623863442184276883178611, 10.88071735527003557479835441964, 10.91650628795341302662702545800
