| L(s) = 1 | + 2.68·2-s − 2.14·3-s + 5.21·4-s − 5.76·6-s − 2.78·7-s + 8.64·8-s + 1.60·9-s − 2.37·11-s − 11.2·12-s + 0.0404·13-s − 7.47·14-s + 12.7·16-s + 1.81·17-s + 4.31·18-s + 5.97·21-s − 6.39·22-s + 2.54·23-s − 18.5·24-s + 0.108·26-s + 2.98·27-s − 14.5·28-s − 2.94·29-s + 2.88·31-s + 17.0·32-s + 5.10·33-s + 4.87·34-s + 8.39·36-s + ⋯ |
| L(s) = 1 | + 1.89·2-s − 1.23·3-s + 2.60·4-s − 2.35·6-s − 1.05·7-s + 3.05·8-s + 0.535·9-s − 0.717·11-s − 3.23·12-s + 0.0112·13-s − 1.99·14-s + 3.19·16-s + 0.440·17-s + 1.01·18-s + 1.30·21-s − 1.36·22-s + 0.530·23-s − 3.78·24-s + 0.0213·26-s + 0.575·27-s − 2.74·28-s − 0.547·29-s + 0.518·31-s + 3.01·32-s + 0.889·33-s + 0.836·34-s + 1.39·36-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9025 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9025 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| 19 | \( 1 \) |
| good | 2 | \( 1 - 2.68T + 2T^{2} \) |
| 3 | \( 1 + 2.14T + 3T^{2} \) |
| 7 | \( 1 + 2.78T + 7T^{2} \) |
| 11 | \( 1 + 2.37T + 11T^{2} \) |
| 13 | \( 1 - 0.0404T + 13T^{2} \) |
| 17 | \( 1 - 1.81T + 17T^{2} \) |
| 23 | \( 1 - 2.54T + 23T^{2} \) |
| 29 | \( 1 + 2.94T + 29T^{2} \) |
| 31 | \( 1 - 2.88T + 31T^{2} \) |
| 37 | \( 1 + 0.227T + 37T^{2} \) |
| 41 | \( 1 + 8.03T + 41T^{2} \) |
| 43 | \( 1 + 5.13T + 43T^{2} \) |
| 47 | \( 1 + 11.0T + 47T^{2} \) |
| 53 | \( 1 - 5.71T + 53T^{2} \) |
| 59 | \( 1 + 11.6T + 59T^{2} \) |
| 61 | \( 1 - 4.58T + 61T^{2} \) |
| 67 | \( 1 - 4.85T + 67T^{2} \) |
| 71 | \( 1 - 7.76T + 71T^{2} \) |
| 73 | \( 1 - 4.24T + 73T^{2} \) |
| 79 | \( 1 + 11.6T + 79T^{2} \) |
| 83 | \( 1 + 4.47T + 83T^{2} \) |
| 89 | \( 1 + 15.3T + 89T^{2} \) |
| 97 | \( 1 - 5.59T + 97T^{2} \) |
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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
−6.84805518552179030414451008195, −6.46577198693175291724136928636, −5.90424575414246735376986908827, −5.14266229047602613327985004017, −4.98347332343818220320895217384, −3.92661009218560619671419081352, −3.22909511560930999012083385587, −2.64079172729150135647999157094, −1.44341427650965967482857774570, 0,
1.44341427650965967482857774570, 2.64079172729150135647999157094, 3.22909511560930999012083385587, 3.92661009218560619671419081352, 4.98347332343818220320895217384, 5.14266229047602613327985004017, 5.90424575414246735376986908827, 6.46577198693175291724136928636, 6.84805518552179030414451008195
