| L(s) = 1 | + 5.96·2-s + 3.52·4-s − 65.6·5-s + 119.·7-s − 169.·8-s − 391.·10-s − 237.·11-s + 999.·13-s + 709.·14-s − 1.12e3·16-s + 1.27e3·17-s + 1.42e3·19-s − 231.·20-s − 1.41e3·22-s − 546.·23-s + 1.18e3·25-s + 5.95e3·26-s + 419.·28-s + 1.54e3·29-s − 7.56e3·31-s − 1.26e3·32-s + 7.60e3·34-s − 7.81e3·35-s − 1.01e3·37-s + 8.50e3·38-s + 1.11e4·40-s − 1.32e4·41-s + ⋯ |
| L(s) = 1 | + 1.05·2-s + 0.110·4-s − 1.17·5-s + 0.918·7-s − 0.937·8-s − 1.23·10-s − 0.592·11-s + 1.63·13-s + 0.967·14-s − 1.09·16-s + 1.07·17-s + 0.906·19-s − 0.129·20-s − 0.623·22-s − 0.215·23-s + 0.378·25-s + 1.72·26-s + 0.101·28-s + 0.341·29-s − 1.41·31-s − 0.219·32-s + 1.12·34-s − 1.07·35-s − 0.121·37-s + 0.955·38-s + 1.10·40-s − 1.23·41-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 | \( 1 - 5.96T + 32T^{2} \) |
| 5 | \( 1 + 65.6T + 3.12e3T^{2} \) |
| 7 | \( 1 - 119.T + 1.68e4T^{2} \) |
| 11 | \( 1 + 237.T + 1.61e5T^{2} \) |
| 13 | \( 1 - 999.T + 3.71e5T^{2} \) |
| 17 | \( 1 - 1.27e3T + 1.41e6T^{2} \) |
| 19 | \( 1 - 1.42e3T + 2.47e6T^{2} \) |
| 23 | \( 1 + 546.T + 6.43e6T^{2} \) |
| 29 | \( 1 - 1.54e3T + 2.05e7T^{2} \) |
| 31 | \( 1 + 7.56e3T + 2.86e7T^{2} \) |
| 37 | \( 1 + 1.01e3T + 6.93e7T^{2} \) |
| 41 | \( 1 + 1.32e4T + 1.15e8T^{2} \) |
| 43 | \( 1 + 2.17e4T + 1.47e8T^{2} \) |
| 47 | \( 1 + 2.30e4T + 2.29e8T^{2} \) |
| 53 | \( 1 + 1.90e4T + 4.18e8T^{2} \) |
| 61 | \( 1 + 7.23e3T + 8.44e8T^{2} \) |
| 67 | \( 1 - 6.98e4T + 1.35e9T^{2} \) |
| 71 | \( 1 + 1.78e4T + 1.80e9T^{2} \) |
| 73 | \( 1 - 9.08e3T + 2.07e9T^{2} \) |
| 79 | \( 1 - 3.14e4T + 3.07e9T^{2} \) |
| 83 | \( 1 - 1.02e5T + 3.93e9T^{2} \) |
| 89 | \( 1 + 1.04e5T + 5.58e9T^{2} \) |
| 97 | \( 1 + 1.17e5T + 8.58e9T^{2} \) |
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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
−9.615229239659583341843345946450, −8.278917486731363225968394955029, −8.069741474534961870683170633525, −6.74219682946457500314092777199, −5.51147662217392572558505924787, −4.90617517329879730328195237608, −3.72345589297365984166641169851, −3.30632919895954239271121348200, −1.44021355161016308718226661750, 0,
1.44021355161016308718226661750, 3.30632919895954239271121348200, 3.72345589297365984166641169851, 4.90617517329879730328195237608, 5.51147662217392572558505924787, 6.74219682946457500314092777199, 8.069741474534961870683170633525, 8.278917486731363225968394955029, 9.615229239659583341843345946450
