| L(s) = 1 | − 0.529·2-s + 3-s − 1.71·4-s + 5-s − 0.529·6-s − 2.25·7-s + 1.97·8-s + 9-s − 0.529·10-s + 0.559·11-s − 1.71·12-s − 13-s + 1.19·14-s + 15-s + 2.39·16-s + 0.271·17-s − 0.529·18-s + 2.13·19-s − 1.71·20-s − 2.25·21-s − 0.296·22-s − 4.48·23-s + 1.97·24-s + 25-s + 0.529·26-s + 27-s + 3.86·28-s + ⋯ |
| L(s) = 1 | − 0.374·2-s + 0.577·3-s − 0.859·4-s + 0.447·5-s − 0.216·6-s − 0.850·7-s + 0.696·8-s + 0.333·9-s − 0.167·10-s + 0.168·11-s − 0.496·12-s − 0.277·13-s + 0.318·14-s + 0.258·15-s + 0.598·16-s + 0.0658·17-s − 0.124·18-s + 0.490·19-s − 0.384·20-s − 0.491·21-s − 0.0632·22-s − 0.934·23-s + 0.402·24-s + 0.200·25-s + 0.103·26-s + 0.192·27-s + 0.731·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6045 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6045 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 - T \) |
| 5 | \( 1 - T \) |
| 13 | \( 1 + T \) |
| 31 | \( 1 + T \) |
| good | 2 | \( 1 + 0.529T + 2T^{2} \) |
| 7 | \( 1 + 2.25T + 7T^{2} \) |
| 11 | \( 1 - 0.559T + 11T^{2} \) |
| 17 | \( 1 - 0.271T + 17T^{2} \) |
| 19 | \( 1 - 2.13T + 19T^{2} \) |
| 23 | \( 1 + 4.48T + 23T^{2} \) |
| 29 | \( 1 - 7.17T + 29T^{2} \) |
| 37 | \( 1 + 7.85T + 37T^{2} \) |
| 41 | \( 1 + 11.4T + 41T^{2} \) |
| 43 | \( 1 + 8.09T + 43T^{2} \) |
| 47 | \( 1 - 2.83T + 47T^{2} \) |
| 53 | \( 1 - 0.326T + 53T^{2} \) |
| 59 | \( 1 - 8.55T + 59T^{2} \) |
| 61 | \( 1 - 5.02T + 61T^{2} \) |
| 67 | \( 1 + 1.31T + 67T^{2} \) |
| 71 | \( 1 - 10.5T + 71T^{2} \) |
| 73 | \( 1 - 8.89T + 73T^{2} \) |
| 79 | \( 1 - 5.70T + 79T^{2} \) |
| 83 | \( 1 - 1.25T + 83T^{2} \) |
| 89 | \( 1 - 15.4T + 89T^{2} \) |
| 97 | \( 1 + 15.9T + 97T^{2} \) |
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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
−8.067657595790720666880100372009, −6.94084490208444809009890859723, −6.55147762368303358391276970068, −5.41242989786093392290986437944, −4.91125860572350896489625192853, −3.83512166530093620735738317827, −3.35096840218116708819367953014, −2.28565128005416089236055237812, −1.26362942153030701964321795314, 0,
1.26362942153030701964321795314, 2.28565128005416089236055237812, 3.35096840218116708819367953014, 3.83512166530093620735738317827, 4.91125860572350896489625192853, 5.41242989786093392290986437944, 6.55147762368303358391276970068, 6.94084490208444809009890859723, 8.067657595790720666880100372009
