| L(s) = 1 | + 8·2-s + 51.0·3-s + 64·4-s + 338.·5-s + 408.·6-s − 143.·7-s + 512·8-s + 420.·9-s + 2.70e3·10-s − 2.23e3·11-s + 3.26e3·12-s + 3.81e3·13-s − 1.14e3·14-s + 1.72e4·15-s + 4.09e3·16-s − 1.99e4·17-s + 3.36e3·18-s + 6.85e3·19-s + 2.16e4·20-s − 7.33e3·21-s − 1.79e4·22-s + 9.77e4·23-s + 2.61e4·24-s + 3.64e4·25-s + 3.05e4·26-s − 9.01e4·27-s − 9.19e3·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1.09·3-s + 0.5·4-s + 1.21·5-s + 0.772·6-s − 0.158·7-s + 0.353·8-s + 0.192·9-s + 0.856·10-s − 0.507·11-s + 0.546·12-s + 0.481·13-s − 0.111·14-s + 1.32·15-s + 0.250·16-s − 0.982·17-s + 0.136·18-s + 0.229·19-s + 0.605·20-s − 0.172·21-s − 0.358·22-s + 1.67·23-s + 0.386·24-s + 0.466·25-s + 0.340·26-s − 0.881·27-s − 0.0791·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 38 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 38 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(4.188024012\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.188024012\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - 8T \) |
| 19 | \( 1 - 6.85e3T \) |
| good | 3 | \( 1 - 51.0T + 2.18e3T^{2} \) |
| 5 | \( 1 - 338.T + 7.81e4T^{2} \) |
| 7 | \( 1 + 143.T + 8.23e5T^{2} \) |
| 11 | \( 1 + 2.23e3T + 1.94e7T^{2} \) |
| 13 | \( 1 - 3.81e3T + 6.27e7T^{2} \) |
| 17 | \( 1 + 1.99e4T + 4.10e8T^{2} \) |
| 23 | \( 1 - 9.77e4T + 3.40e9T^{2} \) |
| 29 | \( 1 - 3.59e4T + 1.72e10T^{2} \) |
| 31 | \( 1 + 1.35e5T + 2.75e10T^{2} \) |
| 37 | \( 1 + 1.12e5T + 9.49e10T^{2} \) |
| 41 | \( 1 + 3.34e5T + 1.94e11T^{2} \) |
| 43 | \( 1 + 7.89e5T + 2.71e11T^{2} \) |
| 47 | \( 1 + 3.45e5T + 5.06e11T^{2} \) |
| 53 | \( 1 + 2.82e5T + 1.17e12T^{2} \) |
| 59 | \( 1 - 1.08e6T + 2.48e12T^{2} \) |
| 61 | \( 1 - 1.60e6T + 3.14e12T^{2} \) |
| 67 | \( 1 - 8.25e5T + 6.06e12T^{2} \) |
| 71 | \( 1 - 1.45e6T + 9.09e12T^{2} \) |
| 73 | \( 1 - 6.35e6T + 1.10e13T^{2} \) |
| 79 | \( 1 + 3.41e6T + 1.92e13T^{2} \) |
| 83 | \( 1 - 2.76e6T + 2.71e13T^{2} \) |
| 89 | \( 1 + 1.95e6T + 4.42e13T^{2} \) |
| 97 | \( 1 + 1.49e7T + 8.07e13T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−14.54680831832613543774524174107, −13.52331762803954829289086070196, −13.05248536667493958351745187368, −11.10712776078351919528325205285, −9.676521142858511169076551990972, −8.504177751150700582470175843934, −6.73800115580524426060838032892, −5.23904749971792351459217545098, −3.22922226701356345782655227584, −1.98206711207948506378978435290,
1.98206711207948506378978435290, 3.22922226701356345782655227584, 5.23904749971792351459217545098, 6.73800115580524426060838032892, 8.504177751150700582470175843934, 9.676521142858511169076551990972, 11.10712776078351919528325205285, 13.05248536667493958351745187368, 13.52331762803954829289086070196, 14.54680831832613543774524174107
