| L(s) = 1 | + 1.61·2-s + 3-s + 0.618·4-s + 5-s + 1.61·6-s + 7-s − 2.23·8-s + 9-s + 1.61·10-s + 11-s + 0.618·12-s − 5.47·13-s + 1.61·14-s + 15-s − 4.85·16-s + 0.763·17-s + 1.61·18-s + 6.70·19-s + 0.618·20-s + 21-s + 1.61·22-s − 7.70·23-s − 2.23·24-s − 4·25-s − 8.85·26-s + 27-s + 0.618·28-s + ⋯ |
| L(s) = 1 | + 1.14·2-s + 0.577·3-s + 0.309·4-s + 0.447·5-s + 0.660·6-s + 0.377·7-s − 0.790·8-s + 0.333·9-s + 0.511·10-s + 0.301·11-s + 0.178·12-s − 1.51·13-s + 0.432·14-s + 0.258·15-s − 1.21·16-s + 0.185·17-s + 0.381·18-s + 1.53·19-s + 0.138·20-s + 0.218·21-s + 0.344·22-s − 1.60·23-s − 0.456·24-s − 0.800·25-s − 1.73·26-s + 0.192·27-s + 0.116·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 231 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 231 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.383631090\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.383631090\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - T \) |
| 7 | \( 1 - T \) |
| 11 | \( 1 - T \) |
| good | 2 | \( 1 - 1.61T + 2T^{2} \) |
| 5 | \( 1 - T + 5T^{2} \) |
| 13 | \( 1 + 5.47T + 13T^{2} \) |
| 17 | \( 1 - 0.763T + 17T^{2} \) |
| 19 | \( 1 - 6.70T + 19T^{2} \) |
| 23 | \( 1 + 7.70T + 23T^{2} \) |
| 29 | \( 1 - 5T + 29T^{2} \) |
| 31 | \( 1 + 0.763T + 31T^{2} \) |
| 37 | \( 1 + 7T + 37T^{2} \) |
| 41 | \( 1 - 6.47T + 41T^{2} \) |
| 43 | \( 1 + 7.70T + 43T^{2} \) |
| 47 | \( 1 + 4.23T + 47T^{2} \) |
| 53 | \( 1 - 10.1T + 53T^{2} \) |
| 59 | \( 1 - 11.1T + 59T^{2} \) |
| 61 | \( 1 - 2T + 61T^{2} \) |
| 67 | \( 1 + 14.2T + 67T^{2} \) |
| 71 | \( 1 - 6.47T + 71T^{2} \) |
| 73 | \( 1 - 13.4T + 73T^{2} \) |
| 79 | \( 1 + 5.52T + 79T^{2} \) |
| 83 | \( 1 - 11.2T + 83T^{2} \) |
| 89 | \( 1 - 4.47T + 89T^{2} \) |
| 97 | \( 1 + 3.70T + 97T^{2} \) |
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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
−12.16162005777975070071729625160, −11.84397984568412955694849699834, −10.06551283195180995532021266754, −9.455778596234711352130152784998, −8.176539531658907131663210060913, −7.04800587197851484422507878598, −5.70314086406454250044583988535, −4.80904691565255811516487049391, −3.60969118437385846600239433648, −2.29149634956536874671352143062,
2.29149634956536874671352143062, 3.60969118437385846600239433648, 4.80904691565255811516487049391, 5.70314086406454250044583988535, 7.04800587197851484422507878598, 8.176539531658907131663210060913, 9.455778596234711352130152784998, 10.06551283195180995532021266754, 11.84397984568412955694849699834, 12.16162005777975070071729625160
