L(s) = 1 | + 3.44·2-s − 116.·4-s + 382.·5-s − 641.·7-s − 840.·8-s + 1.31e3·10-s − 6.57e3·11-s − 2.08e3·13-s − 2.20e3·14-s + 1.19e4·16-s + 3.72e4·17-s − 1.39e4·19-s − 4.43e4·20-s − 2.26e4·22-s + 8.98e4·23-s + 6.79e4·25-s − 7.17e3·26-s + 7.44e4·28-s − 1.07e5·29-s + 2.18e5·31-s + 1.48e5·32-s + 1.28e5·34-s − 2.45e5·35-s − 2.33e5·37-s − 4.80e4·38-s − 3.20e5·40-s + 5.19e5·41-s + ⋯ |
L(s) = 1 | + 0.304·2-s − 0.907·4-s + 1.36·5-s − 0.706·7-s − 0.580·8-s + 0.415·10-s − 1.48·11-s − 0.263·13-s − 0.214·14-s + 0.731·16-s + 1.83·17-s − 0.466·19-s − 1.24·20-s − 0.452·22-s + 1.53·23-s + 0.869·25-s − 0.0800·26-s + 0.641·28-s − 0.817·29-s + 1.31·31-s + 0.802·32-s + 0.558·34-s − 0.966·35-s − 0.758·37-s − 0.141·38-s − 0.793·40-s + 1.17·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 531 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 531 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 \) |
| 59 | \( 1 - 2.05e5T \) |
good | 2 | \( 1 - 3.44T + 128T^{2} \) |
| 5 | \( 1 - 382.T + 7.81e4T^{2} \) |
| 7 | \( 1 + 641.T + 8.23e5T^{2} \) |
| 11 | \( 1 + 6.57e3T + 1.94e7T^{2} \) |
| 13 | \( 1 + 2.08e3T + 6.27e7T^{2} \) |
| 17 | \( 1 - 3.72e4T + 4.10e8T^{2} \) |
| 19 | \( 1 + 1.39e4T + 8.93e8T^{2} \) |
| 23 | \( 1 - 8.98e4T + 3.40e9T^{2} \) |
| 29 | \( 1 + 1.07e5T + 1.72e10T^{2} \) |
| 31 | \( 1 - 2.18e5T + 2.75e10T^{2} \) |
| 37 | \( 1 + 2.33e5T + 9.49e10T^{2} \) |
| 41 | \( 1 - 5.19e5T + 1.94e11T^{2} \) |
| 43 | \( 1 + 9.07e5T + 2.71e11T^{2} \) |
| 47 | \( 1 - 7.59e5T + 5.06e11T^{2} \) |
| 53 | \( 1 + 3.80e5T + 1.17e12T^{2} \) |
| 61 | \( 1 + 2.37e6T + 3.14e12T^{2} \) |
| 67 | \( 1 - 1.91e6T + 6.06e12T^{2} \) |
| 71 | \( 1 - 2.83e6T + 9.09e12T^{2} \) |
| 73 | \( 1 + 4.60e6T + 1.10e13T^{2} \) |
| 79 | \( 1 + 7.37e6T + 1.92e13T^{2} \) |
| 83 | \( 1 - 8.05e6T + 2.71e13T^{2} \) |
| 89 | \( 1 + 5.97e6T + 4.42e13T^{2} \) |
| 97 | \( 1 - 1.56e6T + 8.07e13T^{2} \) |
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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
−9.508258675812378070071370605777, −8.507330545772110374319368828730, −7.48382042759560287696063411674, −6.19539517023542490222874671938, −5.45756468604768846671396831562, −4.88581630587227747121463118881, −3.34496791124621879204006749003, −2.62267260962162363091255841240, −1.17698809274804263402409721326, 0,
1.17698809274804263402409721326, 2.62267260962162363091255841240, 3.34496791124621879204006749003, 4.88581630587227747121463118881, 5.45756468604768846671396831562, 6.19539517023542490222874671938, 7.48382042759560287696063411674, 8.507330545772110374319368828730, 9.508258675812378070071370605777