| L(s) = 1 | − 1.29e5·2-s − 6.82e7·3-s + 8.07e9·4-s − 1.52e11·5-s + 8.81e12·6-s − 1.39e14·7-s + 6.68e13·8-s − 8.95e14·9-s + 1.96e16·10-s − 1.11e17·11-s − 5.51e17·12-s + 3.82e18·13-s + 1.79e19·14-s + 1.04e19·15-s − 7.79e19·16-s + 1.35e20·17-s + 1.15e20·18-s + 1.50e21·19-s − 1.23e21·20-s + 9.51e21·21-s + 1.44e22·22-s + 2.26e22·23-s − 4.56e21·24-s + 2.32e22·25-s − 4.93e23·26-s + 4.40e23·27-s − 1.12e24·28-s + ⋯ |
| L(s) = 1 | − 1.39·2-s − 0.915·3-s + 0.939·4-s − 0.447·5-s + 1.27·6-s − 1.58·7-s + 0.0839·8-s − 0.161·9-s + 0.622·10-s − 0.733·11-s − 0.860·12-s + 1.59·13-s + 2.20·14-s + 0.409·15-s − 1.05·16-s + 0.676·17-s + 0.224·18-s + 1.20·19-s − 0.420·20-s + 1.45·21-s + 1.02·22-s + 0.769·23-s − 0.0769·24-s + 0.200·25-s − 2.22·26-s + 1.06·27-s − 1.48·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(34-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5 ^{s/2} \, \Gamma_{\C}(s+33/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(17)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{35}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + 1.52e11T \) |
| good | 2 | \( 1 + 1.29e5T + 8.58e9T^{2} \) |
| 3 | \( 1 + 6.82e7T + 5.55e15T^{2} \) |
| 7 | \( 1 + 1.39e14T + 7.73e27T^{2} \) |
| 11 | \( 1 + 1.11e17T + 2.32e34T^{2} \) |
| 13 | \( 1 - 3.82e18T + 5.75e36T^{2} \) |
| 17 | \( 1 - 1.35e20T + 4.02e40T^{2} \) |
| 19 | \( 1 - 1.50e21T + 1.58e42T^{2} \) |
| 23 | \( 1 - 2.26e22T + 8.65e44T^{2} \) |
| 29 | \( 1 + 6.82e23T + 1.81e48T^{2} \) |
| 31 | \( 1 + 5.41e24T + 1.64e49T^{2} \) |
| 37 | \( 1 + 7.55e24T + 5.63e51T^{2} \) |
| 41 | \( 1 + 1.95e26T + 1.66e53T^{2} \) |
| 43 | \( 1 - 5.44e26T + 8.02e53T^{2} \) |
| 47 | \( 1 + 3.58e27T + 1.51e55T^{2} \) |
| 53 | \( 1 - 4.28e28T + 7.96e56T^{2} \) |
| 59 | \( 1 + 2.15e28T + 2.74e58T^{2} \) |
| 61 | \( 1 - 5.24e29T + 8.23e58T^{2} \) |
| 67 | \( 1 + 1.47e30T + 1.82e60T^{2} \) |
| 71 | \( 1 + 1.24e30T + 1.23e61T^{2} \) |
| 73 | \( 1 + 9.14e30T + 3.08e61T^{2} \) |
| 79 | \( 1 + 2.52e31T + 4.18e62T^{2} \) |
| 83 | \( 1 + 1.28e31T + 2.13e63T^{2} \) |
| 89 | \( 1 - 1.22e32T + 2.13e64T^{2} \) |
| 97 | \( 1 - 2.05e31T + 3.65e65T^{2} \) |
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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
−16.01417531583702851255545789527, −13.13848939401902070662832303057, −11.38197413170177910578789601639, −10.22107690129088085391143206879, −8.828679055593915378418729351198, −7.18164491004867355002602034511, −5.72581932017110231922042543767, −3.29324260045397388126301025346, −0.945238004110062940083758615377, 0,
0.945238004110062940083758615377, 3.29324260045397388126301025346, 5.72581932017110231922042543767, 7.18164491004867355002602034511, 8.828679055593915378418729351198, 10.22107690129088085391143206879, 11.38197413170177910578789601639, 13.13848939401902070662832303057, 16.01417531583702851255545789527
