| L(s) = 1 | + 1.47·2-s − 0.401·3-s − 5.82·4-s − 11.8·5-s − 0.591·6-s + 11.0·7-s − 20.3·8-s − 26.8·9-s − 17.4·10-s + 32.8·11-s + 2.33·12-s + 87.1·13-s + 16.2·14-s + 4.73·15-s + 16.4·16-s + 33.2·17-s − 39.6·18-s − 6.15·19-s + 68.7·20-s − 4.42·21-s + 48.4·22-s − 151.·23-s + 8.18·24-s + 14.3·25-s + 128.·26-s + 21.5·27-s − 64.1·28-s + ⋯ |
| L(s) = 1 | + 0.521·2-s − 0.0772·3-s − 0.727·4-s − 1.05·5-s − 0.0402·6-s + 0.595·7-s − 0.901·8-s − 0.994·9-s − 0.550·10-s + 0.899·11-s + 0.0561·12-s + 1.86·13-s + 0.310·14-s + 0.0815·15-s + 0.257·16-s + 0.474·17-s − 0.518·18-s − 0.0743·19-s + 0.768·20-s − 0.0459·21-s + 0.469·22-s − 1.37·23-s + 0.0695·24-s + 0.114·25-s + 0.970·26-s + 0.153·27-s − 0.433·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 961 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 961 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 31 | \( 1 \) |
| good | 2 | \( 1 - 1.47T + 8T^{2} \) |
| 3 | \( 1 + 0.401T + 27T^{2} \) |
| 5 | \( 1 + 11.8T + 125T^{2} \) |
| 7 | \( 1 - 11.0T + 343T^{2} \) |
| 11 | \( 1 - 32.8T + 1.33e3T^{2} \) |
| 13 | \( 1 - 87.1T + 2.19e3T^{2} \) |
| 17 | \( 1 - 33.2T + 4.91e3T^{2} \) |
| 19 | \( 1 + 6.15T + 6.85e3T^{2} \) |
| 23 | \( 1 + 151.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 177.T + 2.43e4T^{2} \) |
| 37 | \( 1 + 61.6T + 5.06e4T^{2} \) |
| 41 | \( 1 - 141.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 281.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 79.1T + 1.03e5T^{2} \) |
| 53 | \( 1 + 69.5T + 1.48e5T^{2} \) |
| 59 | \( 1 + 596.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 469.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 855.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 538.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 261.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 957.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 258.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 1.28e3T + 7.04e5T^{2} \) |
| 97 | \( 1 - 889.T + 9.12e5T^{2} \) |
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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
−8.861057309202314548859851666029, −8.476438727818810427734661330976, −7.81278487415893119395855652246, −6.32729715759655562962101830168, −5.77937476534862268847629281644, −4.58611018769740684489211544302, −3.87855244353928490627525462843, −3.19353878408329868228081665878, −1.27937355599541663905802925809, 0,
1.27937355599541663905802925809, 3.19353878408329868228081665878, 3.87855244353928490627525462843, 4.58611018769740684489211544302, 5.77937476534862268847629281644, 6.32729715759655562962101830168, 7.81278487415893119395855652246, 8.476438727818810427734661330976, 8.861057309202314548859851666029
