| L(s) = 1 | − 19.5·2-s + 255.·4-s + 72.4·5-s − 64.7·7-s − 2.48e3·8-s − 1.41e3·10-s + 7.51e3·11-s − 5.97e3·13-s + 1.26e3·14-s + 1.60e4·16-s + 8.37e3·17-s − 5.98e3·19-s + 1.84e4·20-s − 1.47e5·22-s + 2.18e4·23-s − 7.28e4·25-s + 1.16e5·26-s − 1.65e4·28-s + 9.84e3·29-s + 2.42e5·31-s + 4.13e3·32-s − 1.64e5·34-s − 4.69e3·35-s − 8.37e4·37-s + 1.17e5·38-s − 1.80e5·40-s − 3.47e5·41-s + ⋯ |
| L(s) = 1 | − 1.73·2-s + 1.99·4-s + 0.259·5-s − 0.0713·7-s − 1.71·8-s − 0.448·10-s + 1.70·11-s − 0.754·13-s + 0.123·14-s + 0.980·16-s + 0.413·17-s − 0.200·19-s + 0.516·20-s − 2.94·22-s + 0.374·23-s − 0.932·25-s + 1.30·26-s − 0.142·28-s + 0.0749·29-s + 1.46·31-s + 0.0223·32-s − 0.715·34-s − 0.0185·35-s − 0.271·37-s + 0.346·38-s − 0.445·40-s − 0.787·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 531 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 531 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 \) |
| 59 | \( 1 + 2.05e5T \) |
| good | 2 | \( 1 + 19.5T + 128T^{2} \) |
| 5 | \( 1 - 72.4T + 7.81e4T^{2} \) |
| 7 | \( 1 + 64.7T + 8.23e5T^{2} \) |
| 11 | \( 1 - 7.51e3T + 1.94e7T^{2} \) |
| 13 | \( 1 + 5.97e3T + 6.27e7T^{2} \) |
| 17 | \( 1 - 8.37e3T + 4.10e8T^{2} \) |
| 19 | \( 1 + 5.98e3T + 8.93e8T^{2} \) |
| 23 | \( 1 - 2.18e4T + 3.40e9T^{2} \) |
| 29 | \( 1 - 9.84e3T + 1.72e10T^{2} \) |
| 31 | \( 1 - 2.42e5T + 2.75e10T^{2} \) |
| 37 | \( 1 + 8.37e4T + 9.49e10T^{2} \) |
| 41 | \( 1 + 3.47e5T + 1.94e11T^{2} \) |
| 43 | \( 1 - 3.62e5T + 2.71e11T^{2} \) |
| 47 | \( 1 + 7.72e5T + 5.06e11T^{2} \) |
| 53 | \( 1 - 6.90e4T + 1.17e12T^{2} \) |
| 61 | \( 1 + 2.38e6T + 3.14e12T^{2} \) |
| 67 | \( 1 + 2.97e6T + 6.06e12T^{2} \) |
| 71 | \( 1 + 5.08e6T + 9.09e12T^{2} \) |
| 73 | \( 1 - 2.94e6T + 1.10e13T^{2} \) |
| 79 | \( 1 + 4.30e6T + 1.92e13T^{2} \) |
| 83 | \( 1 - 4.28e6T + 2.71e13T^{2} \) |
| 89 | \( 1 - 9.12e6T + 4.42e13T^{2} \) |
| 97 | \( 1 - 8.42e6T + 8.07e13T^{2} \) |
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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
−9.374494961120170879050829235531, −8.542213275853784852674022373180, −7.66202916427820766618390789929, −6.76917775624531951056141999235, −6.10027839158397048175042420338, −4.53113849368316724570678165700, −3.11500997521155473517577170827, −1.87338616550499088216460571852, −1.12731003329409622245840080605, 0,
1.12731003329409622245840080605, 1.87338616550499088216460571852, 3.11500997521155473517577170827, 4.53113849368316724570678165700, 6.10027839158397048175042420338, 6.76917775624531951056141999235, 7.66202916427820766618390789929, 8.542213275853784852674022373180, 9.374494961120170879050829235531
