| L(s) = 1 | − 7.32·2-s + 21.6·4-s − 46.7·5-s − 85.8·7-s + 75.6·8-s + 342.·10-s + 408.·11-s + 27.9·13-s + 628.·14-s − 1.24e3·16-s − 1.58e3·17-s − 2.20e3·19-s − 1.01e3·20-s − 2.99e3·22-s + 3.96e3·23-s − 936.·25-s − 204.·26-s − 1.85e3·28-s + 2.90e3·29-s + 4.85e3·31-s + 6.72e3·32-s + 1.16e4·34-s + 4.01e3·35-s + 2.83e3·37-s + 1.61e4·38-s − 3.54e3·40-s − 3.69e3·41-s + ⋯ |
| L(s) = 1 | − 1.29·2-s + 0.677·4-s − 0.836·5-s − 0.661·7-s + 0.418·8-s + 1.08·10-s + 1.01·11-s + 0.0458·13-s + 0.857·14-s − 1.21·16-s − 1.33·17-s − 1.40·19-s − 0.566·20-s − 1.31·22-s + 1.56·23-s − 0.299·25-s − 0.0593·26-s − 0.448·28-s + 0.641·29-s + 0.907·31-s + 1.16·32-s + 1.72·34-s + 0.553·35-s + 0.340·37-s + 1.81·38-s − 0.349·40-s − 0.343·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 531 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 531 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 59 | \( 1 - 3.48e3T \) |
| good | 2 | \( 1 + 7.32T + 32T^{2} \) |
| 5 | \( 1 + 46.7T + 3.12e3T^{2} \) |
| 7 | \( 1 + 85.8T + 1.68e4T^{2} \) |
| 11 | \( 1 - 408.T + 1.61e5T^{2} \) |
| 13 | \( 1 - 27.9T + 3.71e5T^{2} \) |
| 17 | \( 1 + 1.58e3T + 1.41e6T^{2} \) |
| 19 | \( 1 + 2.20e3T + 2.47e6T^{2} \) |
| 23 | \( 1 - 3.96e3T + 6.43e6T^{2} \) |
| 29 | \( 1 - 2.90e3T + 2.05e7T^{2} \) |
| 31 | \( 1 - 4.85e3T + 2.86e7T^{2} \) |
| 37 | \( 1 - 2.83e3T + 6.93e7T^{2} \) |
| 41 | \( 1 + 3.69e3T + 1.15e8T^{2} \) |
| 43 | \( 1 + 8.79e3T + 1.47e8T^{2} \) |
| 47 | \( 1 - 2.10e4T + 2.29e8T^{2} \) |
| 53 | \( 1 - 1.10e4T + 4.18e8T^{2} \) |
| 61 | \( 1 - 4.53e4T + 8.44e8T^{2} \) |
| 67 | \( 1 + 4.08e4T + 1.35e9T^{2} \) |
| 71 | \( 1 - 2.70e4T + 1.80e9T^{2} \) |
| 73 | \( 1 - 7.04e4T + 2.07e9T^{2} \) |
| 79 | \( 1 - 7.16e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + 9.32e4T + 3.93e9T^{2} \) |
| 89 | \( 1 - 1.16e5T + 5.58e9T^{2} \) |
| 97 | \( 1 - 9.11e4T + 8.58e9T^{2} \) |
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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
−9.387801609123183811193861986725, −8.781020081016162505461535099029, −8.118084361176529759498791846412, −6.90593002436644704585021461430, −6.52724286557562609545017224877, −4.66335582784147849833706784030, −3.83310631182914901065666579024, −2.35300663628110779469252510704, −0.953186654439709059536946736539, 0,
0.953186654439709059536946736539, 2.35300663628110779469252510704, 3.83310631182914901065666579024, 4.66335582784147849833706784030, 6.52724286557562609545017224877, 6.90593002436644704585021461430, 8.118084361176529759498791846412, 8.781020081016162505461535099029, 9.387801609123183811193861986725
