| L(s) = 1 | + 27·3-s − 6·5-s + 316·7-s + 729·9-s − 1.45e3·11-s + 2.19e3·13-s − 162·15-s + 1.47e4·17-s − 1.79e4·19-s + 8.53e3·21-s − 2.30e4·23-s − 7.80e4·25-s + 1.96e4·27-s − 9.08e4·29-s + 1.32e5·31-s − 3.92e4·33-s − 1.89e3·35-s − 2.78e5·37-s + 5.93e4·39-s − 1.66e5·41-s + 6.67e5·43-s − 4.37e3·45-s + 1.42e5·47-s − 7.23e5·49-s + 3.98e5·51-s − 4.93e5·53-s + 8.71e3·55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.0214·5-s + 0.348·7-s + 1/3·9-s − 0.328·11-s + 0.277·13-s − 0.0123·15-s + 0.728·17-s − 0.601·19-s + 0.201·21-s − 0.394·23-s − 0.999·25-s + 0.192·27-s − 0.691·29-s + 0.800·31-s − 0.189·33-s − 0.00747·35-s − 0.903·37-s + 0.160·39-s − 0.378·41-s + 1.27·43-s − 0.00715·45-s + 0.199·47-s − 0.878·49-s + 0.420·51-s − 0.454·53-s + 0.00706·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 624 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 624 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - p^{3} T \) |
| 13 | \( 1 - p^{3} T \) |
| good | 5 | \( 1 + 6 T + p^{7} T^{2} \) |
| 7 | \( 1 - 316 T + p^{7} T^{2} \) |
| 11 | \( 1 + 12 p^{2} T + p^{7} T^{2} \) |
| 17 | \( 1 - 14754 T + p^{7} T^{2} \) |
| 19 | \( 1 + 17984 T + p^{7} T^{2} \) |
| 23 | \( 1 + 23040 T + p^{7} T^{2} \) |
| 29 | \( 1 + 90834 T + p^{7} T^{2} \) |
| 31 | \( 1 - 132772 T + p^{7} T^{2} \) |
| 37 | \( 1 + 278458 T + p^{7} T^{2} \) |
| 41 | \( 1 + 166986 T + p^{7} T^{2} \) |
| 43 | \( 1 - 667108 T + p^{7} T^{2} \) |
| 47 | \( 1 - 142176 T + p^{7} T^{2} \) |
| 53 | \( 1 + 493074 T + p^{7} T^{2} \) |
| 59 | \( 1 + 655620 T + p^{7} T^{2} \) |
| 61 | \( 1 + 463114 T + p^{7} T^{2} \) |
| 67 | \( 1 - 2036632 T + p^{7} T^{2} \) |
| 71 | \( 1 + 1199136 T + p^{7} T^{2} \) |
| 73 | \( 1 + 1546126 T + p^{7} T^{2} \) |
| 79 | \( 1 + 2284568 T + p^{7} T^{2} \) |
| 83 | \( 1 + 1352076 T + p^{7} T^{2} \) |
| 89 | \( 1 - 8194254 T + p^{7} T^{2} \) |
| 97 | \( 1 + 7175926 T + p^{7} T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.002320528947252862574577918926, −8.110992621517586475670564374077, −7.55332441570615840912486702846, −6.38102990566168652375084473158, −5.41745487571532988623657861395, −4.31803833195657884511722778218, −3.41640045736830795683888469880, −2.30459928111698796171157784092, −1.35671002072750490147819314662, 0,
1.35671002072750490147819314662, 2.30459928111698796171157784092, 3.41640045736830795683888469880, 4.31803833195657884511722778218, 5.41745487571532988623657861395, 6.38102990566168652375084473158, 7.55332441570615840912486702846, 8.110992621517586475670564374077, 9.002320528947252862574577918926
