| L(s) = 1 | + 0.0794·2-s − 1.99·4-s − 2.46·5-s − 4.68·7-s − 0.317·8-s − 0.195·10-s − 2.05·11-s − 0.761·13-s − 0.372·14-s + 3.96·16-s + 2.06·17-s + 19-s + 4.91·20-s − 0.163·22-s − 5.89·23-s + 1.07·25-s − 0.0604·26-s + 9.34·28-s + 6.84·29-s + 4.16·31-s + 0.949·32-s + 0.163·34-s + 11.5·35-s + 1.78·37-s + 0.0794·38-s + 0.782·40-s − 3.33·41-s + ⋯ |
| L(s) = 1 | + 0.0562·2-s − 0.996·4-s − 1.10·5-s − 1.77·7-s − 0.112·8-s − 0.0619·10-s − 0.618·11-s − 0.211·13-s − 0.0995·14-s + 0.990·16-s + 0.499·17-s + 0.229·19-s + 1.09·20-s − 0.0347·22-s − 1.23·23-s + 0.214·25-s − 0.0118·26-s + 1.76·28-s + 1.27·29-s + 0.747·31-s + 0.167·32-s + 0.0280·34-s + 1.95·35-s + 0.294·37-s + 0.0128·38-s + 0.123·40-s − 0.520·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8037 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8037 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 19 | \( 1 - T \) |
| 47 | \( 1 - T \) |
| good | 2 | \( 1 - 0.0794T + 2T^{2} \) |
| 5 | \( 1 + 2.46T + 5T^{2} \) |
| 7 | \( 1 + 4.68T + 7T^{2} \) |
| 11 | \( 1 + 2.05T + 11T^{2} \) |
| 13 | \( 1 + 0.761T + 13T^{2} \) |
| 17 | \( 1 - 2.06T + 17T^{2} \) |
| 23 | \( 1 + 5.89T + 23T^{2} \) |
| 29 | \( 1 - 6.84T + 29T^{2} \) |
| 31 | \( 1 - 4.16T + 31T^{2} \) |
| 37 | \( 1 - 1.78T + 37T^{2} \) |
| 41 | \( 1 + 3.33T + 41T^{2} \) |
| 43 | \( 1 + 10.7T + 43T^{2} \) |
| 53 | \( 1 - 4.19T + 53T^{2} \) |
| 59 | \( 1 - 7.20T + 59T^{2} \) |
| 61 | \( 1 + 1.72T + 61T^{2} \) |
| 67 | \( 1 - 11.8T + 67T^{2} \) |
| 71 | \( 1 - 4.40T + 71T^{2} \) |
| 73 | \( 1 - 8.19T + 73T^{2} \) |
| 79 | \( 1 + 5.07T + 79T^{2} \) |
| 83 | \( 1 - 11.3T + 83T^{2} \) |
| 89 | \( 1 + 11.4T + 89T^{2} \) |
| 97 | \( 1 - 2.25T + 97T^{2} \) |
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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
−7.60978521581663492048876848333, −6.77240750261772399822951157522, −6.14035433869670417310490865918, −5.31691773529817882037140349442, −4.57946627152307637541705057426, −3.71470704535539576609106983847, −3.42098239378715365201481155678, −2.51854066689771732114403373338, −0.75829531780837776393407849750, 0,
0.75829531780837776393407849750, 2.51854066689771732114403373338, 3.42098239378715365201481155678, 3.71470704535539576609106983847, 4.57946627152307637541705057426, 5.31691773529817882037140349442, 6.14035433869670417310490865918, 6.77240750261772399822951157522, 7.60978521581663492048876848333
