| L(s) = 1 | − 8·2-s − 27·3-s + 64·4-s + 137.·5-s + 216·6-s + 1.18e3·7-s − 512·8-s + 729·9-s − 1.10e3·10-s + 3.24e3·11-s − 1.72e3·12-s + 4.95e3·13-s − 9.46e3·14-s − 3.71e3·15-s + 4.09e3·16-s − 2.72e4·17-s − 5.83e3·18-s + 1.87e4·19-s + 8.81e3·20-s − 3.19e4·21-s − 2.59e4·22-s − 7.20e4·23-s + 1.38e4·24-s − 5.91e4·25-s − 3.96e4·26-s − 1.96e4·27-s + 7.56e4·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s + 0.5·4-s + 0.492·5-s + 0.408·6-s + 1.30·7-s − 0.353·8-s + 0.333·9-s − 0.348·10-s + 0.735·11-s − 0.288·12-s + 0.625·13-s − 0.921·14-s − 0.284·15-s + 0.250·16-s − 1.34·17-s − 0.235·18-s + 0.627·19-s + 0.246·20-s − 0.752·21-s − 0.519·22-s − 1.23·23-s + 0.204·24-s − 0.757·25-s − 0.442·26-s − 0.192·27-s + 0.651·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 354 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 354 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + 8T \) |
| 3 | \( 1 + 27T \) |
| 59 | \( 1 - 2.05e5T \) |
| good | 5 | \( 1 - 137.T + 7.81e4T^{2} \) |
| 7 | \( 1 - 1.18e3T + 8.23e5T^{2} \) |
| 11 | \( 1 - 3.24e3T + 1.94e7T^{2} \) |
| 13 | \( 1 - 4.95e3T + 6.27e7T^{2} \) |
| 17 | \( 1 + 2.72e4T + 4.10e8T^{2} \) |
| 19 | \( 1 - 1.87e4T + 8.93e8T^{2} \) |
| 23 | \( 1 + 7.20e4T + 3.40e9T^{2} \) |
| 29 | \( 1 + 1.16e5T + 1.72e10T^{2} \) |
| 31 | \( 1 - 1.53e5T + 2.75e10T^{2} \) |
| 37 | \( 1 + 4.25e5T + 9.49e10T^{2} \) |
| 41 | \( 1 + 8.14e5T + 1.94e11T^{2} \) |
| 43 | \( 1 + 4.67e5T + 2.71e11T^{2} \) |
| 47 | \( 1 - 6.29e5T + 5.06e11T^{2} \) |
| 53 | \( 1 - 2.68e5T + 1.17e12T^{2} \) |
| 61 | \( 1 + 2.49e6T + 3.14e12T^{2} \) |
| 67 | \( 1 - 2.28e6T + 6.06e12T^{2} \) |
| 71 | \( 1 + 3.67e5T + 9.09e12T^{2} \) |
| 73 | \( 1 - 3.73e6T + 1.10e13T^{2} \) |
| 79 | \( 1 + 4.20e6T + 1.92e13T^{2} \) |
| 83 | \( 1 + 6.56e5T + 2.71e13T^{2} \) |
| 89 | \( 1 + 3.52e6T + 4.42e13T^{2} \) |
| 97 | \( 1 - 8.05e6T + 8.07e13T^{2} \) |
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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
−9.868539997302065147049404190382, −8.844300199916133733085082354191, −8.084779753327534969751739050261, −6.94466709546455342166047824014, −6.06339477042101918103675604073, −5.03939651011458283908023154808, −3.86017679111105267285470373384, −1.99470300476130074548757153385, −1.40713750645922990659470344286, 0,
1.40713750645922990659470344286, 1.99470300476130074548757153385, 3.86017679111105267285470373384, 5.03939651011458283908023154808, 6.06339477042101918103675604073, 6.94466709546455342166047824014, 8.084779753327534969751739050261, 8.844300199916133733085082354191, 9.868539997302065147049404190382
