| L(s) = 1 | + 6.70·2-s + 4.47·3-s + 29.0·4-s + 25·5-s + 30.0·6-s + 87.2·8-s − 61·9-s + 167.·10-s − 62·11-s + 129.·12-s − 67.0·13-s + 111.·15-s + 121.·16-s − 409.·18-s + 361·19-s + 725.·20-s − 415.·22-s + 390.·24-s + 625·25-s − 450·26-s − 635.·27-s + 750.·30-s − 583.·32-s − 277.·33-s − 1.76e3·36-s − 2.64e3·37-s + 2.42e3·38-s + ⋯ |
| L(s) = 1 | + 1.67·2-s + 0.496·3-s + 1.81·4-s + 5-s + 0.833·6-s + 1.36·8-s − 0.753·9-s + 1.67·10-s − 0.512·11-s + 0.900·12-s − 0.396·13-s + 0.496·15-s + 0.472·16-s − 1.26·18-s + 19-s + 1.81·20-s − 0.859·22-s + 0.677·24-s + 25-s − 0.665·26-s − 0.871·27-s + 0.833·30-s − 0.569·32-s − 0.254·33-s − 1.36·36-s − 1.93·37-s + 1.67·38-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 95 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 95 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(5.154471411\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.154471411\) |
| \(L(3)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 - 25T \) |
| 19 | \( 1 - 361T \) |
| good | 2 | \( 1 - 6.70T + 16T^{2} \) |
| 3 | \( 1 - 4.47T + 81T^{2} \) |
| 7 | \( 1 - 2.40e3T^{2} \) |
| 11 | \( 1 + 62T + 1.46e4T^{2} \) |
| 13 | \( 1 + 67.0T + 2.85e4T^{2} \) |
| 17 | \( 1 - 8.35e4T^{2} \) |
| 23 | \( 1 - 2.79e5T^{2} \) |
| 29 | \( 1 - 7.07e5T^{2} \) |
| 31 | \( 1 - 9.23e5T^{2} \) |
| 37 | \( 1 + 2.64e3T + 1.87e6T^{2} \) |
| 41 | \( 1 - 2.82e6T^{2} \) |
| 43 | \( 1 - 3.41e6T^{2} \) |
| 47 | \( 1 - 4.87e6T^{2} \) |
| 53 | \( 1 - 791.T + 7.89e6T^{2} \) |
| 59 | \( 1 - 1.21e7T^{2} \) |
| 61 | \( 1 - 7.13e3T + 1.38e7T^{2} \) |
| 67 | \( 1 + 6.07e3T + 2.01e7T^{2} \) |
| 71 | \( 1 - 2.54e7T^{2} \) |
| 73 | \( 1 - 2.83e7T^{2} \) |
| 79 | \( 1 - 3.89e7T^{2} \) |
| 83 | \( 1 - 4.74e7T^{2} \) |
| 89 | \( 1 - 6.27e7T^{2} \) |
| 97 | \( 1 + 1.54e4T + 8.85e7T^{2} \) |
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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
−13.57334943631846293688007237545, −12.54643651824025999483925167147, −11.50513753515577700174771054630, −10.21498510190369114511038332773, −8.848879223948621224497156625464, −7.19133250618896566611253659942, −5.83737890371271249716412607421, −5.06408842290981846085975709017, −3.32073381822115920946437634512, −2.25572542021785236939160522503,
2.25572542021785236939160522503, 3.32073381822115920946437634512, 5.06408842290981846085975709017, 5.83737890371271249716412607421, 7.19133250618896566611253659942, 8.848879223948621224497156625464, 10.21498510190369114511038332773, 11.50513753515577700174771054630, 12.54643651824025999483925167147, 13.57334943631846293688007237545
