| L(s) = 1 | − 5.21·2-s − 4.77·4-s + 21.6·5-s + 150.·7-s + 191.·8-s − 113.·10-s − 527.·11-s + 51.3·13-s − 783.·14-s − 848.·16-s + 968.·17-s + 2.27e3·19-s − 103.·20-s + 2.75e3·22-s − 3.80e3·23-s − 2.65e3·25-s − 268.·26-s − 716.·28-s − 7.26e3·29-s − 513.·31-s − 1.71e3·32-s − 5.05e3·34-s + 3.25e3·35-s + 1.13e4·37-s − 1.18e4·38-s + 4.15e3·40-s − 1.70e4·41-s + ⋯ |
| L(s) = 1 | − 0.922·2-s − 0.149·4-s + 0.387·5-s + 1.15·7-s + 1.05·8-s − 0.357·10-s − 1.31·11-s + 0.0843·13-s − 1.06·14-s − 0.828·16-s + 0.812·17-s + 1.44·19-s − 0.0577·20-s + 1.21·22-s − 1.49·23-s − 0.849·25-s − 0.0777·26-s − 0.172·28-s − 1.60·29-s − 0.0958·31-s − 0.295·32-s − 0.749·34-s + 0.448·35-s + 1.35·37-s − 1.33·38-s + 0.410·40-s − 1.58·41-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 | \( 1 + 5.21T + 32T^{2} \) |
| 5 | \( 1 - 21.6T + 3.12e3T^{2} \) |
| 7 | \( 1 - 150.T + 1.68e4T^{2} \) |
| 11 | \( 1 + 527.T + 1.61e5T^{2} \) |
| 13 | \( 1 - 51.3T + 3.71e5T^{2} \) |
| 17 | \( 1 - 968.T + 1.41e6T^{2} \) |
| 19 | \( 1 - 2.27e3T + 2.47e6T^{2} \) |
| 23 | \( 1 + 3.80e3T + 6.43e6T^{2} \) |
| 29 | \( 1 + 7.26e3T + 2.05e7T^{2} \) |
| 31 | \( 1 + 513.T + 2.86e7T^{2} \) |
| 37 | \( 1 - 1.13e4T + 6.93e7T^{2} \) |
| 41 | \( 1 + 1.70e4T + 1.15e8T^{2} \) |
| 43 | \( 1 + 7.35e3T + 1.47e8T^{2} \) |
| 47 | \( 1 - 1.53e4T + 2.29e8T^{2} \) |
| 53 | \( 1 - 3.20e4T + 4.18e8T^{2} \) |
| 61 | \( 1 - 1.38e4T + 8.44e8T^{2} \) |
| 67 | \( 1 + 2.62e4T + 1.35e9T^{2} \) |
| 71 | \( 1 - 4.92e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 5.31e3T + 2.07e9T^{2} \) |
| 79 | \( 1 + 4.11e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + 6.09e4T + 3.93e9T^{2} \) |
| 89 | \( 1 + 4.24e3T + 5.58e9T^{2} \) |
| 97 | \( 1 + 4.91e3T + 8.58e9T^{2} \) |
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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
−9.809966799680744273372106993476, −8.648757295160583829915758379257, −7.77855233686366104394282311784, −7.53862127841705795327803746245, −5.66516691472098941829929213339, −5.11931536261155168625407372301, −3.85171532007094226746757941826, −2.21536575880564621911213458345, −1.27684468367200196016906824524, 0,
1.27684468367200196016906824524, 2.21536575880564621911213458345, 3.85171532007094226746757941826, 5.11931536261155168625407372301, 5.66516691472098941829929213339, 7.53862127841705795327803746245, 7.77855233686366104394282311784, 8.648757295160583829915758379257, 9.809966799680744273372106993476
