| L(s) = 1 | − 2·2-s + 6.22·3-s + 4·4-s − 9.89·5-s − 12.4·6-s + 7·7-s − 8·8-s + 11.7·9-s + 19.7·10-s − 41.5·11-s + 24.9·12-s + 18.7·13-s − 14·14-s − 61.5·15-s + 16·16-s − 19.3·17-s − 23.5·18-s − 52.3·19-s − 39.5·20-s + 43.5·21-s + 83.0·22-s − 23·23-s − 49.8·24-s − 27.1·25-s − 37.4·26-s − 94.8·27-s + 28·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1.19·3-s + 0.5·4-s − 0.884·5-s − 0.847·6-s + 0.377·7-s − 0.353·8-s + 0.435·9-s + 0.625·10-s − 1.13·11-s + 0.599·12-s + 0.399·13-s − 0.267·14-s − 1.05·15-s + 0.250·16-s − 0.275·17-s − 0.308·18-s − 0.631·19-s − 0.442·20-s + 0.452·21-s + 0.804·22-s − 0.208·23-s − 0.423·24-s − 0.217·25-s − 0.282·26-s − 0.676·27-s + 0.188·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 322 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 322 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 + 2T \) |
| 7 | \( 1 - 7T \) |
| 23 | \( 1 + 23T \) |
| good | 3 | \( 1 - 6.22T + 27T^{2} \) |
| 5 | \( 1 + 9.89T + 125T^{2} \) |
| 11 | \( 1 + 41.5T + 1.33e3T^{2} \) |
| 13 | \( 1 - 18.7T + 2.19e3T^{2} \) |
| 17 | \( 1 + 19.3T + 4.91e3T^{2} \) |
| 19 | \( 1 + 52.3T + 6.85e3T^{2} \) |
| 29 | \( 1 + 169.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 191.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 366.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 81.0T + 6.89e4T^{2} \) |
| 43 | \( 1 - 202.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 86.9T + 1.03e5T^{2} \) |
| 53 | \( 1 + 708.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 618.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 258.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 167.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 281.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 643.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 397.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 1.43e3T + 5.71e5T^{2} \) |
| 89 | \( 1 + 497.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 193.T + 9.12e5T^{2} \) |
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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
−10.78215733107329165531589588682, −9.506249200061919314125822343294, −8.739715334958647901547510108959, −7.78756122622661435685464281531, −7.61769800177540783422810974398, −5.91269618559784720465206250691, −4.29108360984605284501629230687, −3.12900740479843562859264024369, −1.98895476068483418595305245526, 0,
1.98895476068483418595305245526, 3.12900740479843562859264024369, 4.29108360984605284501629230687, 5.91269618559784720465206250691, 7.61769800177540783422810974398, 7.78756122622661435685464281531, 8.739715334958647901547510108959, 9.506249200061919314125822343294, 10.78215733107329165531589588682
