| L(s) = 1 | − 2.68·2-s − 24.7·4-s + 83.5·5-s − 48.3·7-s + 152.·8-s − 224.·10-s − 19.9·11-s + 611.·13-s + 130.·14-s + 381.·16-s − 1.52e3·17-s − 1.27e3·19-s − 2.06e3·20-s + 53.7·22-s − 3.30e3·23-s + 3.84e3·25-s − 1.64e3·26-s + 1.19e3·28-s + 2.17e3·29-s + 7.16e3·31-s − 5.91e3·32-s + 4.10e3·34-s − 4.03e3·35-s − 5.90e3·37-s + 3.42e3·38-s + 1.27e4·40-s − 2.58e3·41-s + ⋯ |
| L(s) = 1 | − 0.475·2-s − 0.773·4-s + 1.49·5-s − 0.372·7-s + 0.843·8-s − 0.710·10-s − 0.0498·11-s + 1.00·13-s + 0.177·14-s + 0.373·16-s − 1.28·17-s − 0.808·19-s − 1.15·20-s + 0.0236·22-s − 1.30·23-s + 1.23·25-s − 0.477·26-s + 0.288·28-s + 0.479·29-s + 1.33·31-s − 1.02·32-s + 0.608·34-s − 0.556·35-s − 0.709·37-s + 0.384·38-s + 1.25·40-s − 0.239·41-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 | \( 1 + 2.68T + 32T^{2} \) |
| 5 | \( 1 - 83.5T + 3.12e3T^{2} \) |
| 7 | \( 1 + 48.3T + 1.68e4T^{2} \) |
| 11 | \( 1 + 19.9T + 1.61e5T^{2} \) |
| 13 | \( 1 - 611.T + 3.71e5T^{2} \) |
| 17 | \( 1 + 1.52e3T + 1.41e6T^{2} \) |
| 19 | \( 1 + 1.27e3T + 2.47e6T^{2} \) |
| 23 | \( 1 + 3.30e3T + 6.43e6T^{2} \) |
| 29 | \( 1 - 2.17e3T + 2.05e7T^{2} \) |
| 31 | \( 1 - 7.16e3T + 2.86e7T^{2} \) |
| 37 | \( 1 + 5.90e3T + 6.93e7T^{2} \) |
| 41 | \( 1 + 2.58e3T + 1.15e8T^{2} \) |
| 43 | \( 1 + 5.33e3T + 1.47e8T^{2} \) |
| 47 | \( 1 - 1.05e4T + 2.29e8T^{2} \) |
| 53 | \( 1 + 6.04e3T + 4.18e8T^{2} \) |
| 61 | \( 1 - 134.T + 8.44e8T^{2} \) |
| 67 | \( 1 - 4.98e4T + 1.35e9T^{2} \) |
| 71 | \( 1 + 6.47e4T + 1.80e9T^{2} \) |
| 73 | \( 1 - 4.36e4T + 2.07e9T^{2} \) |
| 79 | \( 1 - 7.48e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + 5.27e3T + 3.93e9T^{2} \) |
| 89 | \( 1 + 8.54e4T + 5.58e9T^{2} \) |
| 97 | \( 1 - 4.73e4T + 8.58e9T^{2} \) |
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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
−9.653236192161493927417968229767, −8.786116928513970843006256101232, −8.223626467685994403401023798547, −6.63561074291282431437839390188, −6.06338181195812731714507872692, −4.94339965533465570836746673428, −3.90522396490704552279917838271, −2.35824938043626103782540135924, −1.35297104165360712046428582204, 0,
1.35297104165360712046428582204, 2.35824938043626103782540135924, 3.90522396490704552279917838271, 4.94339965533465570836746673428, 6.06338181195812731714507872692, 6.63561074291282431437839390188, 8.223626467685994403401023798547, 8.786116928513970843006256101232, 9.653236192161493927417968229767
