| L(s) = 1 | − 2-s + 1.31·3-s + 4-s − 3.25·5-s − 1.31·6-s + 0.670·7-s − 8-s − 1.28·9-s + 3.25·10-s + 11-s + 1.31·12-s − 2.21·13-s − 0.670·14-s − 4.26·15-s + 16-s − 6.10·17-s + 1.28·18-s − 3.25·20-s + 0.879·21-s − 22-s + 8.37·23-s − 1.31·24-s + 5.56·25-s + 2.21·26-s − 5.61·27-s + 0.670·28-s − 1.33·29-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.757·3-s + 0.5·4-s − 1.45·5-s − 0.535·6-s + 0.253·7-s − 0.353·8-s − 0.426·9-s + 1.02·10-s + 0.301·11-s + 0.378·12-s − 0.615·13-s − 0.179·14-s − 1.10·15-s + 0.250·16-s − 1.48·17-s + 0.301·18-s − 0.726·20-s + 0.191·21-s − 0.213·22-s + 1.74·23-s − 0.267·24-s + 1.11·25-s + 0.434·26-s − 1.08·27-s + 0.126·28-s − 0.247·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7942 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7942 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 11 | \( 1 - T \) |
| 19 | \( 1 \) |
| good | 3 | \( 1 - 1.31T + 3T^{2} \) |
| 5 | \( 1 + 3.25T + 5T^{2} \) |
| 7 | \( 1 - 0.670T + 7T^{2} \) |
| 13 | \( 1 + 2.21T + 13T^{2} \) |
| 17 | \( 1 + 6.10T + 17T^{2} \) |
| 23 | \( 1 - 8.37T + 23T^{2} \) |
| 29 | \( 1 + 1.33T + 29T^{2} \) |
| 31 | \( 1 - 9.63T + 31T^{2} \) |
| 37 | \( 1 - 3.75T + 37T^{2} \) |
| 41 | \( 1 + 5.10T + 41T^{2} \) |
| 43 | \( 1 - 10.9T + 43T^{2} \) |
| 47 | \( 1 - 6.15T + 47T^{2} \) |
| 53 | \( 1 - 9.75T + 53T^{2} \) |
| 59 | \( 1 - 9.12T + 59T^{2} \) |
| 61 | \( 1 - 7.49T + 61T^{2} \) |
| 67 | \( 1 + 4.55T + 67T^{2} \) |
| 71 | \( 1 + 2.89T + 71T^{2} \) |
| 73 | \( 1 + 2.93T + 73T^{2} \) |
| 79 | \( 1 + 14.9T + 79T^{2} \) |
| 83 | \( 1 + 2.91T + 83T^{2} \) |
| 89 | \( 1 + 7.36T + 89T^{2} \) |
| 97 | \( 1 + 3.33T + 97T^{2} \) |
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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
−7.52929394737101718352862511592, −7.16335974221616060182654768924, −6.44213955996482390076856728807, −5.33573853860229163583368022572, −4.42512502610555318667942378463, −3.90010341981061542075594397589, −2.82791954941410500375679263264, −2.48899669821496214339104473394, −1.06692589099561376294406775299, 0,
1.06692589099561376294406775299, 2.48899669821496214339104473394, 2.82791954941410500375679263264, 3.90010341981061542075594397589, 4.42512502610555318667942378463, 5.33573853860229163583368022572, 6.44213955996482390076856728807, 7.16335974221616060182654768924, 7.52929394737101718352862511592
