| L(s) = 1 | − 2-s − 1.63·3-s + 4-s − 1.31·5-s + 1.63·6-s + 2.28·7-s − 8-s − 0.316·9-s + 1.31·10-s + 11-s − 1.63·12-s + 3.91·13-s − 2.28·14-s + 2.14·15-s + 16-s − 0.139·17-s + 0.316·18-s − 1.31·20-s − 3.73·21-s − 22-s − 6.62·23-s + 1.63·24-s − 3.27·25-s − 3.91·26-s + 5.43·27-s + 2.28·28-s − 2.81·29-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.945·3-s + 0.5·4-s − 0.586·5-s + 0.668·6-s + 0.862·7-s − 0.353·8-s − 0.105·9-s + 0.414·10-s + 0.301·11-s − 0.472·12-s + 1.08·13-s − 0.609·14-s + 0.555·15-s + 0.250·16-s − 0.0337·17-s + 0.0745·18-s − 0.293·20-s − 0.815·21-s − 0.213·22-s − 1.38·23-s + 0.334·24-s − 0.655·25-s − 0.767·26-s + 1.04·27-s + 0.431·28-s − 0.522·29-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 | \( 1 + 1.63T + 3T^{2} \) |
| 5 | \( 1 + 1.31T + 5T^{2} \) |
| 7 | \( 1 - 2.28T + 7T^{2} \) |
| 13 | \( 1 - 3.91T + 13T^{2} \) |
| 17 | \( 1 + 0.139T + 17T^{2} \) |
| 23 | \( 1 + 6.62T + 23T^{2} \) |
| 29 | \( 1 + 2.81T + 29T^{2} \) |
| 31 | \( 1 - 10.8T + 31T^{2} \) |
| 37 | \( 1 - 10.6T + 37T^{2} \) |
| 41 | \( 1 + 5.77T + 41T^{2} \) |
| 43 | \( 1 + 9.46T + 43T^{2} \) |
| 47 | \( 1 + 4.13T + 47T^{2} \) |
| 53 | \( 1 + 8.62T + 53T^{2} \) |
| 59 | \( 1 - 5.11T + 59T^{2} \) |
| 61 | \( 1 - 0.805T + 61T^{2} \) |
| 67 | \( 1 - 2.73T + 67T^{2} \) |
| 71 | \( 1 + 4.75T + 71T^{2} \) |
| 73 | \( 1 - 14.9T + 73T^{2} \) |
| 79 | \( 1 + 5.82T + 79T^{2} \) |
| 83 | \( 1 + 16.5T + 83T^{2} \) |
| 89 | \( 1 + 4.07T + 89T^{2} \) |
| 97 | \( 1 - 11.7T + 97T^{2} \) |
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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
−7.74006156063045731795209751371, −6.63540874541224760664201143534, −6.26209548426855133438689169295, −5.56542723542019021570151065889, −4.69560674840492347994677684002, −4.01914294759063770998082527783, −3.07830603154245824548315098577, −1.89802697317415071112485647892, −1.05429166311772774474877638524, 0,
1.05429166311772774474877638524, 1.89802697317415071112485647892, 3.07830603154245824548315098577, 4.01914294759063770998082527783, 4.69560674840492347994677684002, 5.56542723542019021570151065889, 6.26209548426855133438689169295, 6.63540874541224760664201143534, 7.74006156063045731795209751371
