| L(s) = 1 | − 8·2-s − 27·3-s + 64·4-s + 125·5-s + 216·6-s − 1.19e3·7-s − 512·8-s + 729·9-s − 1.00e3·10-s + 5.73e3·11-s − 1.72e3·12-s + 2.19e3·13-s + 9.58e3·14-s − 3.37e3·15-s + 4.09e3·16-s − 2.59e4·17-s − 5.83e3·18-s + 3.04e4·19-s + 8.00e3·20-s + 3.23e4·21-s − 4.58e4·22-s − 2.29e4·23-s + 1.38e4·24-s + 1.56e4·25-s − 1.75e4·26-s − 1.96e4·27-s − 7.66e4·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s + 1/2·4-s + 0.447·5-s + 0.408·6-s − 1.32·7-s − 0.353·8-s + 1/3·9-s − 0.316·10-s + 1.29·11-s − 0.288·12-s + 0.277·13-s + 0.933·14-s − 0.258·15-s + 1/4·16-s − 1.28·17-s − 0.235·18-s + 1.01·19-s + 0.223·20-s + 0.762·21-s − 0.918·22-s − 0.393·23-s + 0.204·24-s + 1/5·25-s − 0.196·26-s − 0.192·27-s − 0.660·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 390 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 390 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(1.007622805\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.007622805\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + p^{3} T \) |
| 3 | \( 1 + p^{3} T \) |
| 5 | \( 1 - p^{3} T \) |
| 13 | \( 1 - p^{3} T \) |
| good | 7 | \( 1 + 1198 T + p^{7} T^{2} \) |
| 11 | \( 1 - 5736 T + p^{7} T^{2} \) |
| 17 | \( 1 + 25944 T + p^{7} T^{2} \) |
| 19 | \( 1 - 30458 T + p^{7} T^{2} \) |
| 23 | \( 1 + 22950 T + p^{7} T^{2} \) |
| 29 | \( 1 - 96024 T + p^{7} T^{2} \) |
| 31 | \( 1 + 120424 T + p^{7} T^{2} \) |
| 37 | \( 1 - 218234 T + p^{7} T^{2} \) |
| 41 | \( 1 - 187830 T + p^{7} T^{2} \) |
| 43 | \( 1 + 392932 T + p^{7} T^{2} \) |
| 47 | \( 1 + 336216 T + p^{7} T^{2} \) |
| 53 | \( 1 + 314910 T + p^{7} T^{2} \) |
| 59 | \( 1 + 845544 T + p^{7} T^{2} \) |
| 61 | \( 1 - 2287718 T + p^{7} T^{2} \) |
| 67 | \( 1 - 2475932 T + p^{7} T^{2} \) |
| 71 | \( 1 + 734208 T + p^{7} T^{2} \) |
| 73 | \( 1 + 3666076 T + p^{7} T^{2} \) |
| 79 | \( 1 - 536192 T + p^{7} T^{2} \) |
| 83 | \( 1 + 8986476 T + p^{7} T^{2} \) |
| 89 | \( 1 + 9264054 T + p^{7} T^{2} \) |
| 97 | \( 1 + 6211900 T + p^{7} T^{2} \) |
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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
−9.841396319107187550419010697414, −9.479037303454343475758675931282, −8.530519575776059497536200786489, −7.01715009358433488463653472461, −6.52968314637278400504828228428, −5.71195709582784544916870137357, −4.18275365291347398172661517326, −3.00887004641799497776051127164, −1.63516699173981603851132711812, −0.54386646696353970566519347390,
0.54386646696353970566519347390, 1.63516699173981603851132711812, 3.00887004641799497776051127164, 4.18275365291347398172661517326, 5.71195709582784544916870137357, 6.52968314637278400504828228428, 7.01715009358433488463653472461, 8.530519575776059497536200786489, 9.479037303454343475758675931282, 9.841396319107187550419010697414
