| L(s) = 1 | − 0.689·2-s − 0.902·3-s − 1.52·4-s + 3.70·5-s + 0.622·6-s + 0.763·7-s + 2.43·8-s − 2.18·9-s − 2.55·10-s − 4.11·11-s + 1.37·12-s − 2.90·13-s − 0.526·14-s − 3.34·15-s + 1.37·16-s + 1.30·17-s + 1.50·18-s − 3.79·19-s − 5.65·20-s − 0.688·21-s + 2.83·22-s − 3.28·23-s − 2.19·24-s + 8.74·25-s + 2.00·26-s + 4.67·27-s − 1.16·28-s + ⋯ |
| L(s) = 1 | − 0.487·2-s − 0.521·3-s − 0.762·4-s + 1.65·5-s + 0.254·6-s + 0.288·7-s + 0.859·8-s − 0.728·9-s − 0.808·10-s − 1.24·11-s + 0.397·12-s − 0.807·13-s − 0.140·14-s − 0.863·15-s + 0.343·16-s + 0.317·17-s + 0.355·18-s − 0.871·19-s − 1.26·20-s − 0.150·21-s + 0.604·22-s − 0.684·23-s − 0.447·24-s + 1.74·25-s + 0.393·26-s + 0.900·27-s − 0.219·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 961 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 961 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 31 | \( 1 \) |
| good | 2 | \( 1 + 0.689T + 2T^{2} \) |
| 3 | \( 1 + 0.902T + 3T^{2} \) |
| 5 | \( 1 - 3.70T + 5T^{2} \) |
| 7 | \( 1 - 0.763T + 7T^{2} \) |
| 11 | \( 1 + 4.11T + 11T^{2} \) |
| 13 | \( 1 + 2.90T + 13T^{2} \) |
| 17 | \( 1 - 1.30T + 17T^{2} \) |
| 19 | \( 1 + 3.79T + 19T^{2} \) |
| 23 | \( 1 + 3.28T + 23T^{2} \) |
| 29 | \( 1 - 4.89T + 29T^{2} \) |
| 37 | \( 1 + 10.4T + 37T^{2} \) |
| 41 | \( 1 - 0.755T + 41T^{2} \) |
| 43 | \( 1 + 7.15T + 43T^{2} \) |
| 47 | \( 1 + 0.876T + 47T^{2} \) |
| 53 | \( 1 - 3.57T + 53T^{2} \) |
| 59 | \( 1 - 0.927T + 59T^{2} \) |
| 61 | \( 1 - 2.31T + 61T^{2} \) |
| 67 | \( 1 + 2.08T + 67T^{2} \) |
| 71 | \( 1 + 7.73T + 71T^{2} \) |
| 73 | \( 1 + 5.65T + 73T^{2} \) |
| 79 | \( 1 + 14.0T + 79T^{2} \) |
| 83 | \( 1 + 14.1T + 83T^{2} \) |
| 89 | \( 1 + 4.43T + 89T^{2} \) |
| 97 | \( 1 + 5.27T + 97T^{2} \) |
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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
−9.902547165850844118335709401932, −8.725562600416251022036214881991, −8.273402468220325774017930799560, −7.04583291072696465698228038489, −5.91842195853596301366673049099, −5.31472489550988600660446395121, −4.67819292569890196664591076752, −2.84786392983352421331922844007, −1.74651535124718702871677050199, 0,
1.74651535124718702871677050199, 2.84786392983352421331922844007, 4.67819292569890196664591076752, 5.31472489550988600660446395121, 5.91842195853596301366673049099, 7.04583291072696465698228038489, 8.273402468220325774017930799560, 8.725562600416251022036214881991, 9.902547165850844118335709401932
