| L(s) = 1 | + 1.78e10·2-s + 5.55e15·3-s + 1.71e20·4-s − 1.00e22·5-s + 9.93e25·6-s − 5.50e27·7-s + 4.28e29·8-s + 3.09e31·9-s − 1.79e32·10-s − 7.78e34·11-s + 9.53e35·12-s − 2.25e36·13-s − 9.84e37·14-s − 5.58e37·15-s − 1.76e40·16-s − 9.52e40·17-s + 5.52e41·18-s + 7.47e41·19-s − 1.72e42·20-s − 3.06e43·21-s − 1.39e45·22-s − 3.36e45·23-s + 2.38e45·24-s − 6.76e46·25-s − 4.02e46·26-s + 1.71e47·27-s − 9.45e47·28-s + ⋯ |
| L(s) = 1 | + 1.47·2-s + 0.577·3-s + 1.16·4-s − 0.0385·5-s + 0.849·6-s − 0.269·7-s + 0.239·8-s + 0.333·9-s − 0.0567·10-s − 1.01·11-s + 0.671·12-s − 0.108·13-s − 0.396·14-s − 0.0222·15-s − 0.810·16-s − 0.573·17-s + 0.490·18-s + 0.108·19-s − 0.0448·20-s − 0.155·21-s − 1.48·22-s − 0.812·23-s + 0.138·24-s − 0.998·25-s − 0.159·26-s + 0.192·27-s − 0.313·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(68-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3 ^{s/2} \, \Gamma_{\C}(s+67/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(34)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{69}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - 5.55e15T \) |
| good | 2 | \( 1 - 1.78e10T + 1.47e20T^{2} \) |
| 5 | \( 1 + 1.00e22T + 6.77e46T^{2} \) |
| 7 | \( 1 + 5.50e27T + 4.18e56T^{2} \) |
| 11 | \( 1 + 7.78e34T + 5.93e69T^{2} \) |
| 13 | \( 1 + 2.25e36T + 4.30e74T^{2} \) |
| 17 | \( 1 + 9.52e40T + 2.75e82T^{2} \) |
| 19 | \( 1 - 7.47e41T + 4.74e85T^{2} \) |
| 23 | \( 1 + 3.36e45T + 1.72e91T^{2} \) |
| 29 | \( 1 - 4.41e48T + 9.56e97T^{2} \) |
| 31 | \( 1 - 4.34e49T + 8.34e99T^{2} \) |
| 37 | \( 1 - 1.88e52T + 1.17e105T^{2} \) |
| 41 | \( 1 + 1.15e54T + 1.13e108T^{2} \) |
| 43 | \( 1 + 7.06e54T + 2.76e109T^{2} \) |
| 47 | \( 1 + 6.07e55T + 1.07e112T^{2} \) |
| 53 | \( 1 - 5.87e57T + 3.36e115T^{2} \) |
| 59 | \( 1 - 2.78e59T + 4.43e118T^{2} \) |
| 61 | \( 1 - 5.66e59T + 4.14e119T^{2} \) |
| 67 | \( 1 + 2.08e61T + 2.22e122T^{2} \) |
| 71 | \( 1 - 1.83e62T + 1.08e124T^{2} \) |
| 73 | \( 1 + 4.46e61T + 6.96e124T^{2} \) |
| 79 | \( 1 + 1.25e62T + 1.38e127T^{2} \) |
| 83 | \( 1 + 3.26e64T + 3.78e128T^{2} \) |
| 89 | \( 1 - 2.61e63T + 4.06e130T^{2} \) |
| 97 | \( 1 + 3.95e66T + 1.29e133T^{2} \) |
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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
−13.00146123230544182726159884281, −11.68294227133458598400828829401, −9.964543254462232523395780151055, −8.235677278327620822776490973046, −6.69597220483769356108331354139, −5.36900488061973319724467836792, −4.19682348633033640452727305743, −3.07842111639807273118434886710, −2.08540949229816802454561981239, 0,
2.08540949229816802454561981239, 3.07842111639807273118434886710, 4.19682348633033640452727305743, 5.36900488061973319724467836792, 6.69597220483769356108331354139, 8.235677278327620822776490973046, 9.964543254462232523395780151055, 11.68294227133458598400828829401, 13.00146123230544182726159884281
