| L(s) = 1 | + 0.325·2-s − 1.89·4-s − 3.19·5-s − 2.43·7-s − 1.26·8-s − 1.04·10-s + 11-s − 5.49·13-s − 0.794·14-s + 3.37·16-s − 2.18·17-s + 7.07·19-s + 6.05·20-s + 0.325·22-s + 6.98·23-s + 5.20·25-s − 1.78·26-s + 4.62·28-s + 1.30·29-s + 1.11·31-s + 3.63·32-s − 0.712·34-s + 7.79·35-s + 5.67·37-s + 2.30·38-s + 4.05·40-s − 8.17·41-s + ⋯ |
| L(s) = 1 | + 0.230·2-s − 0.946·4-s − 1.42·5-s − 0.922·7-s − 0.448·8-s − 0.329·10-s + 0.301·11-s − 1.52·13-s − 0.212·14-s + 0.843·16-s − 0.530·17-s + 1.62·19-s + 1.35·20-s + 0.0694·22-s + 1.45·23-s + 1.04·25-s − 0.350·26-s + 0.873·28-s + 0.242·29-s + 0.199·31-s + 0.642·32-s − 0.122·34-s + 1.31·35-s + 0.932·37-s + 0.373·38-s + 0.640·40-s − 1.27·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6039 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6039 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 11 | \( 1 - T \) |
| 61 | \( 1 - T \) |
| good | 2 | \( 1 - 0.325T + 2T^{2} \) |
| 5 | \( 1 + 3.19T + 5T^{2} \) |
| 7 | \( 1 + 2.43T + 7T^{2} \) |
| 13 | \( 1 + 5.49T + 13T^{2} \) |
| 17 | \( 1 + 2.18T + 17T^{2} \) |
| 19 | \( 1 - 7.07T + 19T^{2} \) |
| 23 | \( 1 - 6.98T + 23T^{2} \) |
| 29 | \( 1 - 1.30T + 29T^{2} \) |
| 31 | \( 1 - 1.11T + 31T^{2} \) |
| 37 | \( 1 - 5.67T + 37T^{2} \) |
| 41 | \( 1 + 8.17T + 41T^{2} \) |
| 43 | \( 1 + 6.15T + 43T^{2} \) |
| 47 | \( 1 - 4.94T + 47T^{2} \) |
| 53 | \( 1 - 1.99T + 53T^{2} \) |
| 59 | \( 1 - 5.77T + 59T^{2} \) |
| 67 | \( 1 - 10.3T + 67T^{2} \) |
| 71 | \( 1 + 3.75T + 71T^{2} \) |
| 73 | \( 1 - 8.52T + 73T^{2} \) |
| 79 | \( 1 + 10.5T + 79T^{2} \) |
| 83 | \( 1 + 12.9T + 83T^{2} \) |
| 89 | \( 1 - 5.85T + 89T^{2} \) |
| 97 | \( 1 - 8.17T + 97T^{2} \) |
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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
−7.58676841503494563580609698800, −7.18503152054573094736988714881, −6.41516691762098617020739693349, −5.19952968731570896513452262320, −4.87924262012484424265104662860, −3.99827769447805922425332843149, −3.35298502373627202635420982514, −2.74809379931253759715365464818, −0.892656644318928730283395963131, 0,
0.892656644318928730283395963131, 2.74809379931253759715365464818, 3.35298502373627202635420982514, 3.99827769447805922425332843149, 4.87924262012484424265104662860, 5.19952968731570896513452262320, 6.41516691762098617020739693349, 7.18503152054573094736988714881, 7.58676841503494563580609698800
