| L(s) = 1 | − 2.93·3-s − 2.06·5-s + 5.61·9-s + 6.34·11-s + 0.551·13-s + 6.07·15-s + 1.25·17-s − 2.23·19-s + 23-s − 0.721·25-s − 7.69·27-s − 7.69·29-s − 3.78·31-s − 18.6·33-s + 9.51·37-s − 1.61·39-s − 3.26·41-s − 6.27·43-s − 11.6·45-s − 3.55·47-s − 3.67·51-s + 7.92·53-s − 13.1·55-s + 6.56·57-s + 12.1·59-s + 2.16·61-s − 1.14·65-s + ⋯ |
| L(s) = 1 | − 1.69·3-s − 0.925·5-s + 1.87·9-s + 1.91·11-s + 0.152·13-s + 1.56·15-s + 0.303·17-s − 0.512·19-s + 0.208·23-s − 0.144·25-s − 1.48·27-s − 1.42·29-s − 0.680·31-s − 3.24·33-s + 1.56·37-s − 0.259·39-s − 0.510·41-s − 0.957·43-s − 1.73·45-s − 0.517·47-s − 0.514·51-s + 1.08·53-s − 1.77·55-s + 0.869·57-s + 1.58·59-s + 0.277·61-s − 0.141·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9016 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9016 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 \) |
| 23 | \( 1 - T \) |
| good | 3 | \( 1 + 2.93T + 3T^{2} \) |
| 5 | \( 1 + 2.06T + 5T^{2} \) |
| 11 | \( 1 - 6.34T + 11T^{2} \) |
| 13 | \( 1 - 0.551T + 13T^{2} \) |
| 17 | \( 1 - 1.25T + 17T^{2} \) |
| 19 | \( 1 + 2.23T + 19T^{2} \) |
| 29 | \( 1 + 7.69T + 29T^{2} \) |
| 31 | \( 1 + 3.78T + 31T^{2} \) |
| 37 | \( 1 - 9.51T + 37T^{2} \) |
| 41 | \( 1 + 3.26T + 41T^{2} \) |
| 43 | \( 1 + 6.27T + 43T^{2} \) |
| 47 | \( 1 + 3.55T + 47T^{2} \) |
| 53 | \( 1 - 7.92T + 53T^{2} \) |
| 59 | \( 1 - 12.1T + 59T^{2} \) |
| 61 | \( 1 - 2.16T + 61T^{2} \) |
| 67 | \( 1 + 15.0T + 67T^{2} \) |
| 71 | \( 1 + 7.26T + 71T^{2} \) |
| 73 | \( 1 - 2.94T + 73T^{2} \) |
| 79 | \( 1 - 3.06T + 79T^{2} \) |
| 83 | \( 1 + 1.67T + 83T^{2} \) |
| 89 | \( 1 + 11.1T + 89T^{2} \) |
| 97 | \( 1 + 11.7T + 97T^{2} \) |
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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
−7.06715889901746457713051033284, −6.77165534676177779068524060227, −5.94813785114040990120512401905, −5.52266434122328793128494848478, −4.47124051791684704290952713071, −4.09420316686134327270123354874, −3.42356781199349482030983026328, −1.80759883037622714521823355835, −0.986728265378553159227328724448, 0,
0.986728265378553159227328724448, 1.80759883037622714521823355835, 3.42356781199349482030983026328, 4.09420316686134327270123354874, 4.47124051791684704290952713071, 5.52266434122328793128494848478, 5.94813785114040990120512401905, 6.77165534676177779068524060227, 7.06715889901746457713051033284
