| L(s) = 1 | − 1.77·2-s + 1.16·4-s + 3.42·5-s − 4.73·7-s + 1.48·8-s − 6.10·10-s + 4.00·11-s + 5.24·13-s + 8.42·14-s − 4.97·16-s + 6.72·17-s + 5.62·19-s + 4.00·20-s − 7.13·22-s − 3.38·23-s + 6.76·25-s − 9.33·26-s − 5.52·28-s − 5.64·29-s − 4.15·31-s + 5.88·32-s − 11.9·34-s − 16.2·35-s + 0.459·37-s − 10.0·38-s + 5.07·40-s + 7.63·41-s + ⋯ |
| L(s) = 1 | − 1.25·2-s + 0.583·4-s + 1.53·5-s − 1.78·7-s + 0.523·8-s − 1.93·10-s + 1.20·11-s + 1.45·13-s + 2.25·14-s − 1.24·16-s + 1.63·17-s + 1.29·19-s + 0.895·20-s − 1.52·22-s − 0.705·23-s + 1.35·25-s − 1.83·26-s − 1.04·28-s − 1.04·29-s − 0.745·31-s + 1.04·32-s − 2.05·34-s − 2.74·35-s + 0.0754·37-s − 1.62·38-s + 0.803·40-s + 1.19·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2151 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2151 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.196406581\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.196406581\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 239 | \( 1 + T \) |
| good | 2 | \( 1 + 1.77T + 2T^{2} \) |
| 5 | \( 1 - 3.42T + 5T^{2} \) |
| 7 | \( 1 + 4.73T + 7T^{2} \) |
| 11 | \( 1 - 4.00T + 11T^{2} \) |
| 13 | \( 1 - 5.24T + 13T^{2} \) |
| 17 | \( 1 - 6.72T + 17T^{2} \) |
| 19 | \( 1 - 5.62T + 19T^{2} \) |
| 23 | \( 1 + 3.38T + 23T^{2} \) |
| 29 | \( 1 + 5.64T + 29T^{2} \) |
| 31 | \( 1 + 4.15T + 31T^{2} \) |
| 37 | \( 1 - 0.459T + 37T^{2} \) |
| 41 | \( 1 - 7.63T + 41T^{2} \) |
| 43 | \( 1 + 11.1T + 43T^{2} \) |
| 47 | \( 1 - 9.77T + 47T^{2} \) |
| 53 | \( 1 + 2.85T + 53T^{2} \) |
| 59 | \( 1 + 2.31T + 59T^{2} \) |
| 61 | \( 1 - 6.70T + 61T^{2} \) |
| 67 | \( 1 + 4.26T + 67T^{2} \) |
| 71 | \( 1 - 2.74T + 71T^{2} \) |
| 73 | \( 1 + 5.07T + 73T^{2} \) |
| 79 | \( 1 + 13.2T + 79T^{2} \) |
| 83 | \( 1 + 6.60T + 83T^{2} \) |
| 89 | \( 1 - 7.24T + 89T^{2} \) |
| 97 | \( 1 + 6.52T + 97T^{2} \) |
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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
−9.228032389577990588570984739502, −8.736856071320234182570603225628, −7.53832493196114366070228730283, −6.80183058267795718435244600744, −5.96068665624765129017424556741, −5.65555757096111353788335788213, −3.87796720614642898954879137720, −3.14559131217520459105817176402, −1.72328941813793801322348937611, −0.946707644886169419163471101830,
0.946707644886169419163471101830, 1.72328941813793801322348937611, 3.14559131217520459105817176402, 3.87796720614642898954879137720, 5.65555757096111353788335788213, 5.96068665624765129017424556741, 6.80183058267795718435244600744, 7.53832493196114366070228730283, 8.736856071320234182570603225628, 9.228032389577990588570984739502
