| L(s) = 1 | − 2-s − 2.97·3-s + 4-s + 5-s + 2.97·6-s − 1.89·7-s − 8-s + 5.84·9-s − 10-s + 11-s − 2.97·12-s − 4.17·13-s + 1.89·14-s − 2.97·15-s + 16-s + 0.0330·17-s − 5.84·18-s + 1.87·19-s + 20-s + 5.64·21-s − 22-s − 3.17·23-s + 2.97·24-s + 25-s + 4.17·26-s − 8.46·27-s − 1.89·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1.71·3-s + 0.5·4-s + 0.447·5-s + 1.21·6-s − 0.716·7-s − 0.353·8-s + 1.94·9-s − 0.316·10-s + 0.301·11-s − 0.858·12-s − 1.15·13-s + 0.506·14-s − 0.768·15-s + 0.250·16-s + 0.00801·17-s − 1.37·18-s + 0.431·19-s + 0.223·20-s + 1.23·21-s − 0.213·22-s − 0.662·23-s + 0.607·24-s + 0.200·25-s + 0.818·26-s − 1.62·27-s − 0.358·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8030 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8030 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 11 | \( 1 - T \) |
| 73 | \( 1 + T \) |
| good | 3 | \( 1 + 2.97T + 3T^{2} \) |
| 7 | \( 1 + 1.89T + 7T^{2} \) |
| 13 | \( 1 + 4.17T + 13T^{2} \) |
| 17 | \( 1 - 0.0330T + 17T^{2} \) |
| 19 | \( 1 - 1.87T + 19T^{2} \) |
| 23 | \( 1 + 3.17T + 23T^{2} \) |
| 29 | \( 1 - 4.44T + 29T^{2} \) |
| 31 | \( 1 + 1.09T + 31T^{2} \) |
| 37 | \( 1 + 2.75T + 37T^{2} \) |
| 41 | \( 1 + 4.75T + 41T^{2} \) |
| 43 | \( 1 - 8.12T + 43T^{2} \) |
| 47 | \( 1 - 11.0T + 47T^{2} \) |
| 53 | \( 1 + 9.27T + 53T^{2} \) |
| 59 | \( 1 + 7.08T + 59T^{2} \) |
| 61 | \( 1 + 1.29T + 61T^{2} \) |
| 67 | \( 1 - 8.20T + 67T^{2} \) |
| 71 | \( 1 - 9.05T + 71T^{2} \) |
| 79 | \( 1 - 10.9T + 79T^{2} \) |
| 83 | \( 1 + 11.4T + 83T^{2} \) |
| 89 | \( 1 - 7.96T + 89T^{2} \) |
| 97 | \( 1 + 18.1T + 97T^{2} \) |
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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
−7.28930862470357500706296648481, −6.67680246310737290415462582584, −6.23893329905767473536706933935, −5.53010867778316775097249135320, −4.94515572580891937784986341897, −4.07397641826154149491122670097, −2.94037791217137025179015010909, −1.92948897212113603693013777650, −0.895516997988804701707758955031, 0,
0.895516997988804701707758955031, 1.92948897212113603693013777650, 2.94037791217137025179015010909, 4.07397641826154149491122670097, 4.94515572580891937784986341897, 5.53010867778316775097249135320, 6.23893329905767473536706933935, 6.67680246310737290415462582584, 7.28930862470357500706296648481
