| L(s) = 1 | + 2.16·2-s − 3-s + 2.69·4-s + 2.49·5-s − 2.16·6-s + 1.25·7-s + 1.49·8-s + 9-s + 5.39·10-s + 0.689·11-s − 2.69·12-s + 4.68·13-s + 2.70·14-s − 2.49·15-s − 2.13·16-s − 0.185·17-s + 2.16·18-s + 7.71·19-s + 6.70·20-s − 1.25·21-s + 1.49·22-s − 8.34·23-s − 1.49·24-s + 1.20·25-s + 10.1·26-s − 27-s + 3.36·28-s + ⋯ |
| L(s) = 1 | + 1.53·2-s − 0.577·3-s + 1.34·4-s + 1.11·5-s − 0.884·6-s + 0.472·7-s + 0.529·8-s + 0.333·9-s + 1.70·10-s + 0.207·11-s − 0.776·12-s + 1.29·13-s + 0.724·14-s − 0.643·15-s − 0.534·16-s − 0.0449·17-s + 0.510·18-s + 1.76·19-s + 1.49·20-s − 0.273·21-s + 0.318·22-s − 1.74·23-s − 0.305·24-s + 0.240·25-s + 1.99·26-s − 0.192·27-s + 0.636·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2883 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2883 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(5.009035844\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.009035844\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + T \) |
| 31 | \( 1 \) |
| good | 2 | \( 1 - 2.16T + 2T^{2} \) |
| 5 | \( 1 - 2.49T + 5T^{2} \) |
| 7 | \( 1 - 1.25T + 7T^{2} \) |
| 11 | \( 1 - 0.689T + 11T^{2} \) |
| 13 | \( 1 - 4.68T + 13T^{2} \) |
| 17 | \( 1 + 0.185T + 17T^{2} \) |
| 19 | \( 1 - 7.71T + 19T^{2} \) |
| 23 | \( 1 + 8.34T + 23T^{2} \) |
| 29 | \( 1 - 4.42T + 29T^{2} \) |
| 37 | \( 1 + 1.94T + 37T^{2} \) |
| 41 | \( 1 - 8.55T + 41T^{2} \) |
| 43 | \( 1 + 10.7T + 43T^{2} \) |
| 47 | \( 1 - 4.65T + 47T^{2} \) |
| 53 | \( 1 - 8.40T + 53T^{2} \) |
| 59 | \( 1 - 5.78T + 59T^{2} \) |
| 61 | \( 1 - 3.09T + 61T^{2} \) |
| 67 | \( 1 + 10.0T + 67T^{2} \) |
| 71 | \( 1 + 9.39T + 71T^{2} \) |
| 73 | \( 1 - 11.2T + 73T^{2} \) |
| 79 | \( 1 - 0.210T + 79T^{2} \) |
| 83 | \( 1 - 7.23T + 83T^{2} \) |
| 89 | \( 1 - 2.40T + 89T^{2} \) |
| 97 | \( 1 + 3.68T + 97T^{2} \) |
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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
−8.803508272288703926670092239905, −7.81453464812959107875052923315, −6.79118134470060568369285540920, −6.09105727792506961803912609329, −5.68343676877863871250712423260, −5.04456159758939254770957178394, −4.13660922201056815470269062957, −3.37534038857995484568520128780, −2.23141037042745796452355891792, −1.27889022298535464463104557442,
1.27889022298535464463104557442, 2.23141037042745796452355891792, 3.37534038857995484568520128780, 4.13660922201056815470269062957, 5.04456159758939254770957178394, 5.68343676877863871250712423260, 6.09105727792506961803912609329, 6.79118134470060568369285540920, 7.81453464812959107875052923315, 8.803508272288703926670092239905
