| L(s) = 1 | + 17.9·2-s + 195.·4-s + 98.9·5-s + 159.·7-s + 1.21e3·8-s + 1.77e3·10-s + 4.88e3·11-s + 1.16e4·13-s + 2.86e3·14-s − 3.15e3·16-s + 1.04e4·17-s + 1.49e4·19-s + 1.93e4·20-s + 8.78e4·22-s + 88.0·23-s − 6.83e4·25-s + 2.09e5·26-s + 3.11e4·28-s + 1.02e5·29-s − 2.87e5·31-s − 2.12e5·32-s + 1.88e5·34-s + 1.57e4·35-s + 5.66e5·37-s + 2.69e5·38-s + 1.20e5·40-s − 2.96e5·41-s + ⋯ |
| L(s) = 1 | + 1.59·2-s + 1.52·4-s + 0.353·5-s + 0.175·7-s + 0.839·8-s + 0.562·10-s + 1.10·11-s + 1.47·13-s + 0.278·14-s − 0.192·16-s + 0.516·17-s + 0.500·19-s + 0.540·20-s + 1.75·22-s + 0.00150·23-s − 0.874·25-s + 2.34·26-s + 0.268·28-s + 0.781·29-s − 1.73·31-s − 1.14·32-s + 0.821·34-s + 0.0620·35-s + 1.83·37-s + 0.795·38-s + 0.297·40-s − 0.670·41-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
| \(L(4)\) |
\(\approx\) |
\(8.128689551\) |
| \(L(\frac12)\) |
\(\approx\) |
\(8.128689551\) |
| \(L(\frac{9}{2})\) |
|
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 | \( 1 - 17.9T + 128T^{2} \) |
| 5 | \( 1 - 98.9T + 7.81e4T^{2} \) |
| 7 | \( 1 - 159.T + 8.23e5T^{2} \) |
| 11 | \( 1 - 4.88e3T + 1.94e7T^{2} \) |
| 13 | \( 1 - 1.16e4T + 6.27e7T^{2} \) |
| 17 | \( 1 - 1.04e4T + 4.10e8T^{2} \) |
| 19 | \( 1 - 1.49e4T + 8.93e8T^{2} \) |
| 23 | \( 1 - 88.0T + 3.40e9T^{2} \) |
| 29 | \( 1 - 1.02e5T + 1.72e10T^{2} \) |
| 31 | \( 1 + 2.87e5T + 2.75e10T^{2} \) |
| 37 | \( 1 - 5.66e5T + 9.49e10T^{2} \) |
| 41 | \( 1 + 2.96e5T + 1.94e11T^{2} \) |
| 43 | \( 1 - 7.40e5T + 2.71e11T^{2} \) |
| 47 | \( 1 + 9.58e5T + 5.06e11T^{2} \) |
| 53 | \( 1 - 7.32e5T + 1.17e12T^{2} \) |
| 61 | \( 1 - 2.28e6T + 3.14e12T^{2} \) |
| 67 | \( 1 + 4.29e6T + 6.06e12T^{2} \) |
| 71 | \( 1 - 4.99e6T + 9.09e12T^{2} \) |
| 73 | \( 1 - 3.66e6T + 1.10e13T^{2} \) |
| 79 | \( 1 - 3.18e6T + 1.92e13T^{2} \) |
| 83 | \( 1 - 4.43e6T + 2.71e13T^{2} \) |
| 89 | \( 1 - 1.07e7T + 4.42e13T^{2} \) |
| 97 | \( 1 + 4.05e6T + 8.07e13T^{2} \) |
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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
−9.738107700895182740562364206484, −8.863823483787141570710266547408, −7.66448918570122578727218252444, −6.45813307897639255383093128406, −5.99270874743642502768405507096, −5.05170859244512996145459493333, −3.93323710185731197166532501292, −3.42044707644478569262811868662, −2.07631149802986628972260898954, −1.04153867641568880216658717569,
1.04153867641568880216658717569, 2.07631149802986628972260898954, 3.42044707644478569262811868662, 3.93323710185731197166532501292, 5.05170859244512996145459493333, 5.99270874743642502768405507096, 6.45813307897639255383093128406, 7.66448918570122578727218252444, 8.863823483787141570710266547408, 9.738107700895182740562364206484
