| L(s) = 1 | + 0.249·2-s − 2.30·3-s − 1.93·4-s − 0.574·6-s + 3.96·7-s − 0.983·8-s + 2.29·9-s + 3.13·11-s + 4.45·12-s + 2.65·13-s + 0.989·14-s + 3.62·16-s + 2.25·17-s + 0.573·18-s − 9.11·21-s + 0.782·22-s − 7.58·23-s + 2.26·24-s + 0.664·26-s + 1.61·27-s − 7.67·28-s − 1.36·29-s − 0.894·31-s + 2.87·32-s − 7.21·33-s + 0.563·34-s − 4.45·36-s + ⋯ |
| L(s) = 1 | + 0.176·2-s − 1.32·3-s − 0.968·4-s − 0.234·6-s + 1.49·7-s − 0.347·8-s + 0.765·9-s + 0.945·11-s + 1.28·12-s + 0.737·13-s + 0.264·14-s + 0.907·16-s + 0.547·17-s + 0.135·18-s − 1.98·21-s + 0.166·22-s − 1.58·23-s + 0.462·24-s + 0.130·26-s + 0.311·27-s − 1.45·28-s − 0.253·29-s − 0.160·31-s + 0.507·32-s − 1.25·33-s + 0.0966·34-s − 0.741·36-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9025 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9025 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 5 | \( 1 \) |
| 19 | \( 1 \) |
| good | 2 | \( 1 - 0.249T + 2T^{2} \) |
| 3 | \( 1 + 2.30T + 3T^{2} \) |
| 7 | \( 1 - 3.96T + 7T^{2} \) |
| 11 | \( 1 - 3.13T + 11T^{2} \) |
| 13 | \( 1 - 2.65T + 13T^{2} \) |
| 17 | \( 1 - 2.25T + 17T^{2} \) |
| 23 | \( 1 + 7.58T + 23T^{2} \) |
| 29 | \( 1 + 1.36T + 29T^{2} \) |
| 31 | \( 1 + 0.894T + 31T^{2} \) |
| 37 | \( 1 + 6.62T + 37T^{2} \) |
| 41 | \( 1 + 6.23T + 41T^{2} \) |
| 43 | \( 1 + 1.77T + 43T^{2} \) |
| 47 | \( 1 - 0.176T + 47T^{2} \) |
| 53 | \( 1 + 6.77T + 53T^{2} \) |
| 59 | \( 1 - 7.81T + 59T^{2} \) |
| 61 | \( 1 + 1.03T + 61T^{2} \) |
| 67 | \( 1 + 15.2T + 67T^{2} \) |
| 71 | \( 1 + 10.4T + 71T^{2} \) |
| 73 | \( 1 - 4.18T + 73T^{2} \) |
| 79 | \( 1 + 11.2T + 79T^{2} \) |
| 83 | \( 1 - 13.2T + 83T^{2} \) |
| 89 | \( 1 + 1.48T + 89T^{2} \) |
| 97 | \( 1 - 15.3T + 97T^{2} \) |
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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
−7.40714137771883080619268247073, −6.42420436515286958268386415888, −5.87228168725249005247376114297, −5.30532579262202241495874163952, −4.71417856006586021186944222050, −4.11107657098712565216313465905, −3.41034980581381201782549625736, −1.75647955402892268364566032571, −1.16166203576160460951675106834, 0,
1.16166203576160460951675106834, 1.75647955402892268364566032571, 3.41034980581381201782549625736, 4.11107657098712565216313465905, 4.71417856006586021186944222050, 5.30532579262202241495874163952, 5.87228168725249005247376114297, 6.42420436515286958268386415888, 7.40714137771883080619268247073
