| L(s) = 1 | − 11.7·2-s − 43.4·3-s + 9.96·4-s + 389.·5-s + 510.·6-s + 1.38e3·8-s − 296.·9-s − 4.57e3·10-s + 1.33e3·11-s − 433.·12-s + 3.84e3·13-s − 1.69e4·15-s − 1.75e4·16-s − 2.41e4·17-s + 3.48e3·18-s − 5.45e3·19-s + 3.88e3·20-s − 1.56e4·22-s − 6.39e4·23-s − 6.02e4·24-s + 7.39e4·25-s − 4.51e4·26-s + 1.07e5·27-s + 1.78e5·29-s + 1.99e5·30-s + 1.85e5·31-s + 2.88e4·32-s + ⋯ |
| L(s) = 1 | − 1.03·2-s − 0.929·3-s + 0.0778·4-s + 1.39·5-s + 0.965·6-s + 0.957·8-s − 0.135·9-s − 1.44·10-s + 0.301·11-s − 0.0723·12-s + 0.484·13-s − 1.29·15-s − 1.07·16-s − 1.19·17-s + 0.140·18-s − 0.182·19-s + 0.108·20-s − 0.313·22-s − 1.09·23-s − 0.890·24-s + 0.946·25-s − 0.503·26-s + 1.05·27-s + 1.35·29-s + 1.34·30-s + 1.11·31-s + 0.155·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 539 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 539 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(0.8584458060\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8584458060\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 \) |
| 11 | \( 1 - 1.33e3T \) |
| good | 2 | \( 1 + 11.7T + 128T^{2} \) |
| 3 | \( 1 + 43.4T + 2.18e3T^{2} \) |
| 5 | \( 1 - 389.T + 7.81e4T^{2} \) |
| 13 | \( 1 - 3.84e3T + 6.27e7T^{2} \) |
| 17 | \( 1 + 2.41e4T + 4.10e8T^{2} \) |
| 19 | \( 1 + 5.45e3T + 8.93e8T^{2} \) |
| 23 | \( 1 + 6.39e4T + 3.40e9T^{2} \) |
| 29 | \( 1 - 1.78e5T + 1.72e10T^{2} \) |
| 31 | \( 1 - 1.85e5T + 2.75e10T^{2} \) |
| 37 | \( 1 + 4.09e5T + 9.49e10T^{2} \) |
| 41 | \( 1 - 6.75e5T + 1.94e11T^{2} \) |
| 43 | \( 1 - 3.89e4T + 2.71e11T^{2} \) |
| 47 | \( 1 + 9.49e5T + 5.06e11T^{2} \) |
| 53 | \( 1 - 2.94e5T + 1.17e12T^{2} \) |
| 59 | \( 1 - 8.78e4T + 2.48e12T^{2} \) |
| 61 | \( 1 - 2.78e6T + 3.14e12T^{2} \) |
| 67 | \( 1 - 2.95e6T + 6.06e12T^{2} \) |
| 71 | \( 1 + 4.08e6T + 9.09e12T^{2} \) |
| 73 | \( 1 + 1.95e6T + 1.10e13T^{2} \) |
| 79 | \( 1 + 6.08e5T + 1.92e13T^{2} \) |
| 83 | \( 1 + 2.14e5T + 2.71e13T^{2} \) |
| 89 | \( 1 - 8.30e6T + 4.42e13T^{2} \) |
| 97 | \( 1 - 1.38e6T + 8.07e13T^{2} \) |
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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
−9.788536519765423266431069588546, −8.859046955737133193034974604601, −8.259676396281221210547482420216, −6.74403417378233929911400520969, −6.22836057370081050380151404294, −5.25432328943705442087172969189, −4.29683881860688404386381021681, −2.46473724347136917457290077604, −1.47753333455928911402068860968, −0.52563726521101199961116635039,
0.52563726521101199961116635039, 1.47753333455928911402068860968, 2.46473724347136917457290077604, 4.29683881860688404386381021681, 5.25432328943705442087172969189, 6.22836057370081050380151404294, 6.74403417378233929911400520969, 8.259676396281221210547482420216, 8.859046955737133193034974604601, 9.788536519765423266431069588546
