| L(s) = 1 | − 1.95·2-s + 0.350·3-s + 1.83·4-s − 0.686·6-s + 1.14·7-s + 0.318·8-s − 2.87·9-s + 0.395·11-s + 0.643·12-s − 0.595·13-s − 2.25·14-s − 4.29·16-s − 7.79·17-s + 5.63·18-s + 3.25·19-s + 0.402·21-s − 0.774·22-s + 3.49·23-s + 0.111·24-s + 1.16·26-s − 2.05·27-s + 2.11·28-s + 6.23·29-s − 9.01·31-s + 7.78·32-s + 0.138·33-s + 15.2·34-s + ⋯ |
| L(s) = 1 | − 1.38·2-s + 0.202·3-s + 0.918·4-s − 0.280·6-s + 0.434·7-s + 0.112·8-s − 0.959·9-s + 0.119·11-s + 0.185·12-s − 0.165·13-s − 0.601·14-s − 1.07·16-s − 1.88·17-s + 1.32·18-s + 0.746·19-s + 0.0878·21-s − 0.165·22-s + 0.729·23-s + 0.0228·24-s + 0.228·26-s − 0.396·27-s + 0.398·28-s + 1.15·29-s − 1.61·31-s + 1.37·32-s + 0.0241·33-s + 2.61·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6025 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6025 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 241 | \( 1 - T \) |
| good | 2 | \( 1 + 1.95T + 2T^{2} \) |
| 3 | \( 1 - 0.350T + 3T^{2} \) |
| 7 | \( 1 - 1.14T + 7T^{2} \) |
| 11 | \( 1 - 0.395T + 11T^{2} \) |
| 13 | \( 1 + 0.595T + 13T^{2} \) |
| 17 | \( 1 + 7.79T + 17T^{2} \) |
| 19 | \( 1 - 3.25T + 19T^{2} \) |
| 23 | \( 1 - 3.49T + 23T^{2} \) |
| 29 | \( 1 - 6.23T + 29T^{2} \) |
| 31 | \( 1 + 9.01T + 31T^{2} \) |
| 37 | \( 1 - 0.667T + 37T^{2} \) |
| 41 | \( 1 - 6.05T + 41T^{2} \) |
| 43 | \( 1 - 9.26T + 43T^{2} \) |
| 47 | \( 1 - 13.6T + 47T^{2} \) |
| 53 | \( 1 - 0.719T + 53T^{2} \) |
| 59 | \( 1 + 10.8T + 59T^{2} \) |
| 61 | \( 1 - 2.84T + 61T^{2} \) |
| 67 | \( 1 - 1.17T + 67T^{2} \) |
| 71 | \( 1 + 14.0T + 71T^{2} \) |
| 73 | \( 1 + 4.89T + 73T^{2} \) |
| 79 | \( 1 - 13.1T + 79T^{2} \) |
| 83 | \( 1 - 5.86T + 83T^{2} \) |
| 89 | \( 1 + 11.3T + 89T^{2} \) |
| 97 | \( 1 - 0.175T + 97T^{2} \) |
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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
−7.81171201515408972309841832360, −7.33113022177172149726804589371, −6.57654131161945059437880583741, −5.71154714674878836269980125836, −4.80988061072240494324622250920, −4.08260799986181477868457216642, −2.80609016031972207749190931404, −2.19228729560544830724831016844, −1.10411883823313101568396456016, 0,
1.10411883823313101568396456016, 2.19228729560544830724831016844, 2.80609016031972207749190931404, 4.08260799986181477868457216642, 4.80988061072240494324622250920, 5.71154714674878836269980125836, 6.57654131161945059437880583741, 7.33113022177172149726804589371, 7.81171201515408972309841832360
