| L(s) = 1 | + 2-s − 2.63·3-s + 4-s + 1.62·5-s − 2.63·6-s + 2.40·7-s + 8-s + 3.92·9-s + 1.62·10-s − 3.88·11-s − 2.63·12-s − 0.435·13-s + 2.40·14-s − 4.27·15-s + 16-s − 1.47·17-s + 3.92·18-s − 4.69·19-s + 1.62·20-s − 6.33·21-s − 3.88·22-s + 6.93·23-s − 2.63·24-s − 2.36·25-s − 0.435·26-s − 2.43·27-s + 2.40·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 1.51·3-s + 0.5·4-s + 0.726·5-s − 1.07·6-s + 0.909·7-s + 0.353·8-s + 1.30·9-s + 0.513·10-s − 1.17·11-s − 0.759·12-s − 0.120·13-s + 0.643·14-s − 1.10·15-s + 0.250·16-s − 0.357·17-s + 0.925·18-s − 1.07·19-s + 0.363·20-s − 1.38·21-s − 0.829·22-s + 1.44·23-s − 0.537·24-s − 0.472·25-s − 0.0854·26-s − 0.468·27-s + 0.454·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4034 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4034 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 2017 | \( 1 - T \) |
| good | 3 | \( 1 + 2.63T + 3T^{2} \) |
| 5 | \( 1 - 1.62T + 5T^{2} \) |
| 7 | \( 1 - 2.40T + 7T^{2} \) |
| 11 | \( 1 + 3.88T + 11T^{2} \) |
| 13 | \( 1 + 0.435T + 13T^{2} \) |
| 17 | \( 1 + 1.47T + 17T^{2} \) |
| 19 | \( 1 + 4.69T + 19T^{2} \) |
| 23 | \( 1 - 6.93T + 23T^{2} \) |
| 29 | \( 1 + 0.515T + 29T^{2} \) |
| 31 | \( 1 + 0.199T + 31T^{2} \) |
| 37 | \( 1 + 11.2T + 37T^{2} \) |
| 41 | \( 1 + 11.0T + 41T^{2} \) |
| 43 | \( 1 + 7.50T + 43T^{2} \) |
| 47 | \( 1 + 7.60T + 47T^{2} \) |
| 53 | \( 1 + 3.42T + 53T^{2} \) |
| 59 | \( 1 - 7.97T + 59T^{2} \) |
| 61 | \( 1 - 10.4T + 61T^{2} \) |
| 67 | \( 1 - 1.39T + 67T^{2} \) |
| 71 | \( 1 - 9.93T + 71T^{2} \) |
| 73 | \( 1 + 0.642T + 73T^{2} \) |
| 79 | \( 1 - 14.4T + 79T^{2} \) |
| 83 | \( 1 - 5.55T + 83T^{2} \) |
| 89 | \( 1 + 14.8T + 89T^{2} \) |
| 97 | \( 1 - 7.29T + 97T^{2} \) |
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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
−7.990756898367779173398650113988, −6.79808107290739111458655575325, −6.64180151305362159630865196207, −5.54612647047056073756931118079, −5.06260931968118552249562639579, −4.86212153727624735064752329702, −3.58085170378408161798207575066, −2.30972254379411684216549673111, −1.52864011444204435140453243287, 0,
1.52864011444204435140453243287, 2.30972254379411684216549673111, 3.58085170378408161798207575066, 4.86212153727624735064752329702, 5.06260931968118552249562639579, 5.54612647047056073756931118079, 6.64180151305362159630865196207, 6.79808107290739111458655575325, 7.990756898367779173398650113988
