| L(s) = 1 | + 10.4·2-s − 18.3·4-s − 512.·5-s + 1.73e3·7-s − 1.53e3·8-s − 5.36e3·10-s − 2.05e3·11-s + 3.69e3·13-s + 1.81e4·14-s − 1.36e4·16-s + 3.93e3·17-s − 1.44e4·19-s + 9.41e3·20-s − 2.15e4·22-s − 2.33e4·23-s + 1.84e5·25-s + 3.87e4·26-s − 3.18e4·28-s + 7.48e4·29-s + 2.76e5·31-s + 5.28e4·32-s + 4.11e4·34-s − 8.87e5·35-s − 2.05e4·37-s − 1.51e5·38-s + 7.85e5·40-s + 2.92e5·41-s + ⋯ |
| L(s) = 1 | + 0.925·2-s − 0.143·4-s − 1.83·5-s + 1.90·7-s − 1.05·8-s − 1.69·10-s − 0.465·11-s + 0.466·13-s + 1.76·14-s − 0.835·16-s + 0.194·17-s − 0.483·19-s + 0.263·20-s − 0.431·22-s − 0.399·23-s + 2.35·25-s + 0.431·26-s − 0.274·28-s + 0.569·29-s + 1.66·31-s + 0.284·32-s + 0.179·34-s − 3.49·35-s − 0.0667·37-s − 0.447·38-s + 1.93·40-s + 0.663·41-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 | \( 1 - 10.4T + 128T^{2} \) |
| 5 | \( 1 + 512.T + 7.81e4T^{2} \) |
| 7 | \( 1 - 1.73e3T + 8.23e5T^{2} \) |
| 11 | \( 1 + 2.05e3T + 1.94e7T^{2} \) |
| 13 | \( 1 - 3.69e3T + 6.27e7T^{2} \) |
| 17 | \( 1 - 3.93e3T + 4.10e8T^{2} \) |
| 19 | \( 1 + 1.44e4T + 8.93e8T^{2} \) |
| 23 | \( 1 + 2.33e4T + 3.40e9T^{2} \) |
| 29 | \( 1 - 7.48e4T + 1.72e10T^{2} \) |
| 31 | \( 1 - 2.76e5T + 2.75e10T^{2} \) |
| 37 | \( 1 + 2.05e4T + 9.49e10T^{2} \) |
| 41 | \( 1 - 2.92e5T + 1.94e11T^{2} \) |
| 43 | \( 1 - 1.17e5T + 2.71e11T^{2} \) |
| 47 | \( 1 + 1.08e6T + 5.06e11T^{2} \) |
| 53 | \( 1 + 8.79e5T + 1.17e12T^{2} \) |
| 61 | \( 1 + 3.24e6T + 3.14e12T^{2} \) |
| 67 | \( 1 + 1.01e6T + 6.06e12T^{2} \) |
| 71 | \( 1 + 7.65e5T + 9.09e12T^{2} \) |
| 73 | \( 1 + 3.70e5T + 1.10e13T^{2} \) |
| 79 | \( 1 - 5.68e6T + 1.92e13T^{2} \) |
| 83 | \( 1 - 3.65e5T + 2.71e13T^{2} \) |
| 89 | \( 1 - 9.85e6T + 4.42e13T^{2} \) |
| 97 | \( 1 + 8.40e6T + 8.07e13T^{2} \) |
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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
−8.891014830981505276420790269164, −8.060122140349369458800113801181, −7.84483871707228115079679907066, −6.39758549741867219687487125483, −5.04131509652496002676539409277, −4.56265237450795799606294222704, −3.87584433462958081249554429204, −2.76345785297663946292065049696, −1.15336826713645535538158008896, 0,
1.15336826713645535538158008896, 2.76345785297663946292065049696, 3.87584433462958081249554429204, 4.56265237450795799606294222704, 5.04131509652496002676539409277, 6.39758549741867219687487125483, 7.84483871707228115079679907066, 8.060122140349369458800113801181, 8.891014830981505276420790269164
