| L(s) = 1 | − 8·2-s + 27·3-s + 64·4-s − 195.·5-s − 216·6-s + 306.·7-s − 512·8-s + 729·9-s + 1.56e3·10-s + 4.34e3·11-s + 1.72e3·12-s + 3.58e3·13-s − 2.45e3·14-s − 5.28e3·15-s + 4.09e3·16-s − 3.11e4·17-s − 5.83e3·18-s − 3.40e4·19-s − 1.25e4·20-s + 8.28e3·21-s − 3.47e4·22-s + 3.14e4·23-s − 1.38e4·24-s − 3.97e4·25-s − 2.86e4·26-s + 1.96e4·27-s + 1.96e4·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.577·3-s + 0.5·4-s − 0.700·5-s − 0.408·6-s + 0.338·7-s − 0.353·8-s + 0.333·9-s + 0.495·10-s + 0.984·11-s + 0.288·12-s + 0.452·13-s − 0.239·14-s − 0.404·15-s + 0.250·16-s − 1.53·17-s − 0.235·18-s − 1.13·19-s − 0.350·20-s + 0.195·21-s − 0.695·22-s + 0.538·23-s − 0.204·24-s − 0.508·25-s − 0.319·26-s + 0.192·27-s + 0.169·28-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 | \( 1 + 195.T + 7.81e4T^{2} \) |
| 7 | \( 1 - 306.T + 8.23e5T^{2} \) |
| 11 | \( 1 - 4.34e3T + 1.94e7T^{2} \) |
| 13 | \( 1 - 3.58e3T + 6.27e7T^{2} \) |
| 17 | \( 1 + 3.11e4T + 4.10e8T^{2} \) |
| 19 | \( 1 + 3.40e4T + 8.93e8T^{2} \) |
| 23 | \( 1 - 3.14e4T + 3.40e9T^{2} \) |
| 29 | \( 1 - 1.66e5T + 1.72e10T^{2} \) |
| 31 | \( 1 - 7.56e4T + 2.75e10T^{2} \) |
| 37 | \( 1 + 3.61e4T + 9.49e10T^{2} \) |
| 41 | \( 1 + 4.30e5T + 1.94e11T^{2} \) |
| 43 | \( 1 + 3.47e5T + 2.71e11T^{2} \) |
| 47 | \( 1 - 6.73e5T + 5.06e11T^{2} \) |
| 53 | \( 1 - 4.03e5T + 1.17e12T^{2} \) |
| 61 | \( 1 - 3.11e5T + 3.14e12T^{2} \) |
| 67 | \( 1 + 1.18e6T + 6.06e12T^{2} \) |
| 71 | \( 1 - 5.03e6T + 9.09e12T^{2} \) |
| 73 | \( 1 - 2.72e6T + 1.10e13T^{2} \) |
| 79 | \( 1 + 5.81e5T + 1.92e13T^{2} \) |
| 83 | \( 1 + 6.98e6T + 2.71e13T^{2} \) |
| 89 | \( 1 + 7.43e6T + 4.42e13T^{2} \) |
| 97 | \( 1 + 1.54e7T + 8.07e13T^{2} \) |
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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
−9.642292335850683606813107307783, −8.565475137498779500122186318192, −8.376822121275117888563874737271, −7.03289709226293473050226345391, −6.39027156594339265757503134713, −4.59958130364801107305080550014, −3.73188141498250060381067070899, −2.39800009669369716647050393469, −1.30394201877407684963316962333, 0,
1.30394201877407684963316962333, 2.39800009669369716647050393469, 3.73188141498250060381067070899, 4.59958130364801107305080550014, 6.39027156594339265757503134713, 7.03289709226293473050226345391, 8.376822121275117888563874737271, 8.565475137498779500122186318192, 9.642292335850683606813107307783
