| L(s) = 1 | − 0.452·2-s + 0.461·3-s − 1.79·4-s − 0.208·6-s + 1.59·7-s + 1.71·8-s − 2.78·9-s − 0.580·11-s − 0.828·12-s + 4.75·13-s − 0.723·14-s + 2.81·16-s − 6.34·17-s + 1.26·18-s + 6.21·19-s + 0.737·21-s + 0.262·22-s − 5.12·23-s + 0.792·24-s − 2.15·26-s − 2.67·27-s − 2.87·28-s − 2.57·29-s + 8.34·31-s − 4.70·32-s − 0.268·33-s + 2.87·34-s + ⋯ |
| L(s) = 1 | − 0.319·2-s + 0.266·3-s − 0.897·4-s − 0.0852·6-s + 0.604·7-s + 0.607·8-s − 0.929·9-s − 0.175·11-s − 0.239·12-s + 1.31·13-s − 0.193·14-s + 0.703·16-s − 1.53·17-s + 0.297·18-s + 1.42·19-s + 0.161·21-s + 0.0560·22-s − 1.06·23-s + 0.161·24-s − 0.421·26-s − 0.513·27-s − 0.542·28-s − 0.478·29-s + 1.49·31-s − 0.832·32-s − 0.0466·33-s + 0.492·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1175 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1175 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.216133039\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.216133039\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| 47 | \( 1 + T \) |
| good | 2 | \( 1 + 0.452T + 2T^{2} \) |
| 3 | \( 1 - 0.461T + 3T^{2} \) |
| 7 | \( 1 - 1.59T + 7T^{2} \) |
| 11 | \( 1 + 0.580T + 11T^{2} \) |
| 13 | \( 1 - 4.75T + 13T^{2} \) |
| 17 | \( 1 + 6.34T + 17T^{2} \) |
| 19 | \( 1 - 6.21T + 19T^{2} \) |
| 23 | \( 1 + 5.12T + 23T^{2} \) |
| 29 | \( 1 + 2.57T + 29T^{2} \) |
| 31 | \( 1 - 8.34T + 31T^{2} \) |
| 37 | \( 1 - 7.48T + 37T^{2} \) |
| 41 | \( 1 - 10.9T + 41T^{2} \) |
| 43 | \( 1 - 5.38T + 43T^{2} \) |
| 53 | \( 1 - 6.26T + 53T^{2} \) |
| 59 | \( 1 - 6.74T + 59T^{2} \) |
| 61 | \( 1 - 2.24T + 61T^{2} \) |
| 67 | \( 1 - 7.71T + 67T^{2} \) |
| 71 | \( 1 + 4.87T + 71T^{2} \) |
| 73 | \( 1 + 11.6T + 73T^{2} \) |
| 79 | \( 1 - 16.8T + 79T^{2} \) |
| 83 | \( 1 - 13.1T + 83T^{2} \) |
| 89 | \( 1 - 16.0T + 89T^{2} \) |
| 97 | \( 1 + 1.71T + 97T^{2} \) |
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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
−9.481861900724692714617150781478, −8.977344555565111583603793005291, −8.163373624908406561690872888112, −7.76680147553815386730027316510, −6.31190588234116358781215856484, −5.53070672793871986918842176792, −4.51753757662981142266725541046, −3.70353680786449459339037704127, −2.40143946742436528670721666981, −0.886687638792742920546842192548,
0.886687638792742920546842192548, 2.40143946742436528670721666981, 3.70353680786449459339037704127, 4.51753757662981142266725541046, 5.53070672793871986918842176792, 6.31190588234116358781215856484, 7.76680147553815386730027316510, 8.163373624908406561690872888112, 8.977344555565111583603793005291, 9.481861900724692714617150781478
