| L(s) = 1 | − 1.30·2-s + 2.06·3-s − 0.290·4-s − 5-s − 2.69·6-s + 2.99·8-s + 1.25·9-s + 1.30·10-s − 0.909·11-s − 0.599·12-s + 3.77·13-s − 2.06·15-s − 3.33·16-s − 6.78·17-s − 1.64·18-s + 2.90·19-s + 0.290·20-s + 1.18·22-s − 5.92·23-s + 6.17·24-s + 25-s − 4.93·26-s − 3.59·27-s + 9.84·29-s + 2.69·30-s + 1.67·31-s − 1.62·32-s + ⋯ |
| L(s) = 1 | − 0.924·2-s + 1.19·3-s − 0.145·4-s − 0.447·5-s − 1.10·6-s + 1.05·8-s + 0.419·9-s + 0.413·10-s − 0.274·11-s − 0.172·12-s + 1.04·13-s − 0.532·15-s − 0.833·16-s − 1.64·17-s − 0.387·18-s + 0.666·19-s + 0.0649·20-s + 0.253·22-s − 1.23·23-s + 1.26·24-s + 0.200·25-s − 0.968·26-s − 0.691·27-s + 1.82·29-s + 0.492·30-s + 0.299·31-s − 0.287·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9065 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9065 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 7 | \( 1 \) |
| 37 | \( 1 + T \) |
| good | 2 | \( 1 + 1.30T + 2T^{2} \) |
| 3 | \( 1 - 2.06T + 3T^{2} \) |
| 11 | \( 1 + 0.909T + 11T^{2} \) |
| 13 | \( 1 - 3.77T + 13T^{2} \) |
| 17 | \( 1 + 6.78T + 17T^{2} \) |
| 19 | \( 1 - 2.90T + 19T^{2} \) |
| 23 | \( 1 + 5.92T + 23T^{2} \) |
| 29 | \( 1 - 9.84T + 29T^{2} \) |
| 31 | \( 1 - 1.67T + 31T^{2} \) |
| 41 | \( 1 + 1.01T + 41T^{2} \) |
| 43 | \( 1 + 3.34T + 43T^{2} \) |
| 47 | \( 1 - 4.56T + 47T^{2} \) |
| 53 | \( 1 + 3.37T + 53T^{2} \) |
| 59 | \( 1 - 13.6T + 59T^{2} \) |
| 61 | \( 1 + 4.68T + 61T^{2} \) |
| 67 | \( 1 - 4.73T + 67T^{2} \) |
| 71 | \( 1 - 7.05T + 71T^{2} \) |
| 73 | \( 1 + 13.7T + 73T^{2} \) |
| 79 | \( 1 + 5.83T + 79T^{2} \) |
| 83 | \( 1 + 0.103T + 83T^{2} \) |
| 89 | \( 1 + 0.502T + 89T^{2} \) |
| 97 | \( 1 + 18.6T + 97T^{2} \) |
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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
−7.77295152765217024075367375492, −6.97579966865374360457042941614, −6.28924196798311292148109052031, −5.21058924283140275426694742315, −4.30098388440427314725632746454, −3.87699399636140303562839445948, −2.91352235129729873188224411678, −2.15145241825526964834861459157, −1.18615145222985411771965921524, 0,
1.18615145222985411771965921524, 2.15145241825526964834861459157, 2.91352235129729873188224411678, 3.87699399636140303562839445948, 4.30098388440427314725632746454, 5.21058924283140275426694742315, 6.28924196798311292148109052031, 6.97579966865374360457042941614, 7.77295152765217024075367375492
