| L(s) = 1 | − 2-s + 3.15·3-s + 4-s − 5-s − 3.15·6-s − 0.474·7-s − 8-s + 6.94·9-s + 10-s − 1.05·11-s + 3.15·12-s − 3.43·13-s + 0.474·14-s − 3.15·15-s + 16-s − 4.69·17-s − 6.94·18-s − 6.53·19-s − 20-s − 1.49·21-s + 1.05·22-s − 6.00·23-s − 3.15·24-s + 25-s + 3.43·26-s + 12.4·27-s − 0.474·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1.82·3-s + 0.5·4-s − 0.447·5-s − 1.28·6-s − 0.179·7-s − 0.353·8-s + 2.31·9-s + 0.316·10-s − 0.317·11-s + 0.910·12-s − 0.953·13-s + 0.126·14-s − 0.814·15-s + 0.250·16-s − 1.13·17-s − 1.63·18-s − 1.49·19-s − 0.223·20-s − 0.326·21-s + 0.224·22-s − 1.25·23-s − 0.643·24-s + 0.200·25-s + 0.674·26-s + 2.39·27-s − 0.0896·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4010 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4010 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 + T \) |
| 5 | \( 1 + T \) |
| 401 | \( 1 + T \) |
| good | 3 | \( 1 - 3.15T + 3T^{2} \) |
| 7 | \( 1 + 0.474T + 7T^{2} \) |
| 11 | \( 1 + 1.05T + 11T^{2} \) |
| 13 | \( 1 + 3.43T + 13T^{2} \) |
| 17 | \( 1 + 4.69T + 17T^{2} \) |
| 19 | \( 1 + 6.53T + 19T^{2} \) |
| 23 | \( 1 + 6.00T + 23T^{2} \) |
| 29 | \( 1 - 8.30T + 29T^{2} \) |
| 31 | \( 1 - 0.592T + 31T^{2} \) |
| 37 | \( 1 - 0.575T + 37T^{2} \) |
| 41 | \( 1 - 2.13T + 41T^{2} \) |
| 43 | \( 1 + 3.03T + 43T^{2} \) |
| 47 | \( 1 + 1.96T + 47T^{2} \) |
| 53 | \( 1 + 6.30T + 53T^{2} \) |
| 59 | \( 1 + 4.62T + 59T^{2} \) |
| 61 | \( 1 - 13.0T + 61T^{2} \) |
| 67 | \( 1 + 14.9T + 67T^{2} \) |
| 71 | \( 1 + 10.7T + 71T^{2} \) |
| 73 | \( 1 + 6.62T + 73T^{2} \) |
| 79 | \( 1 + 10.7T + 79T^{2} \) |
| 83 | \( 1 - 12.6T + 83T^{2} \) |
| 89 | \( 1 - 2.47T + 89T^{2} \) |
| 97 | \( 1 + 7.44T + 97T^{2} \) |
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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
−8.205176098652513788995128843087, −7.68537943133778383248273803915, −6.90147921598437379445543909267, −6.28435891466954120568655332072, −4.65036546030646392683856974540, −4.18106735630310969842956073417, −3.08272135971167313871694288390, −2.46401785391805847781229686618, −1.75024833826189812282127020893, 0,
1.75024833826189812282127020893, 2.46401785391805847781229686618, 3.08272135971167313871694288390, 4.18106735630310969842956073417, 4.65036546030646392683856974540, 6.28435891466954120568655332072, 6.90147921598437379445543909267, 7.68537943133778383248273803915, 8.205176098652513788995128843087
