| L(s) = 1 | + 10.7·2-s − 32.3·3-s − 12.0·4-s − 347.·6-s − 420.·7-s − 1.50e3·8-s − 1.14e3·9-s − 6.82e3·11-s + 387.·12-s + 1.01e4·13-s − 4.52e3·14-s − 1.47e4·16-s + 1.56e4·17-s − 1.23e4·18-s − 6.86e3·19-s + 1.35e4·21-s − 7.35e4·22-s − 2.92e4·23-s + 4.87e4·24-s + 1.09e5·26-s + 1.07e5·27-s + 5.04e3·28-s − 2.55e4·29-s + 8.21e4·31-s + 3.46e4·32-s + 2.20e5·33-s + 1.68e5·34-s + ⋯ |
| L(s) = 1 | + 0.951·2-s − 0.690·3-s − 0.0937·4-s − 0.657·6-s − 0.462·7-s − 1.04·8-s − 0.522·9-s − 1.54·11-s + 0.0647·12-s + 1.28·13-s − 0.440·14-s − 0.897·16-s + 0.774·17-s − 0.497·18-s − 0.229·19-s + 0.319·21-s − 1.47·22-s − 0.500·23-s + 0.719·24-s + 1.21·26-s + 1.05·27-s + 0.0433·28-s − 0.194·29-s + 0.495·31-s + 0.186·32-s + 1.06·33-s + 0.736·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 25 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 25 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| good | 2 | \( 1 - 10.7T + 128T^{2} \) |
| 3 | \( 1 + 32.3T + 2.18e3T^{2} \) |
| 7 | \( 1 + 420.T + 8.23e5T^{2} \) |
| 11 | \( 1 + 6.82e3T + 1.94e7T^{2} \) |
| 13 | \( 1 - 1.01e4T + 6.27e7T^{2} \) |
| 17 | \( 1 - 1.56e4T + 4.10e8T^{2} \) |
| 19 | \( 1 + 6.86e3T + 8.93e8T^{2} \) |
| 23 | \( 1 + 2.92e4T + 3.40e9T^{2} \) |
| 29 | \( 1 + 2.55e4T + 1.72e10T^{2} \) |
| 31 | \( 1 - 8.21e4T + 2.75e10T^{2} \) |
| 37 | \( 1 + 2.23e5T + 9.49e10T^{2} \) |
| 41 | \( 1 + 5.33e5T + 1.94e11T^{2} \) |
| 43 | \( 1 + 7.08e5T + 2.71e11T^{2} \) |
| 47 | \( 1 - 5.82e3T + 5.06e11T^{2} \) |
| 53 | \( 1 - 5.89e5T + 1.17e12T^{2} \) |
| 59 | \( 1 + 1.43e6T + 2.48e12T^{2} \) |
| 61 | \( 1 - 1.38e6T + 3.14e12T^{2} \) |
| 67 | \( 1 + 2.71e6T + 6.06e12T^{2} \) |
| 71 | \( 1 + 4.81e5T + 9.09e12T^{2} \) |
| 73 | \( 1 + 1.48e6T + 1.10e13T^{2} \) |
| 79 | \( 1 - 1.05e6T + 1.92e13T^{2} \) |
| 83 | \( 1 - 2.60e6T + 2.71e13T^{2} \) |
| 89 | \( 1 + 5.64e6T + 4.42e13T^{2} \) |
| 97 | \( 1 - 1.20e7T + 8.07e13T^{2} \) |
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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
−15.33232123919012325496933600721, −13.84481076462811257914444539010, −12.94567428917971928457347779358, −11.74114402062395342599699503693, −10.32251569245436240376323332977, −8.434667252026757798950217634537, −6.17079253694754335447791025510, −5.15746859765721955026539699123, −3.25772023830105731930052251909, 0,
3.25772023830105731930052251909, 5.15746859765721955026539699123, 6.17079253694754335447791025510, 8.434667252026757798950217634537, 10.32251569245436240376323332977, 11.74114402062395342599699503693, 12.94567428917971928457347779358, 13.84481076462811257914444539010, 15.33232123919012325496933600721
