| L(s) = 1 | − 11.3·2-s + 85.1·3-s − 0.0696·4-s + 523.·5-s − 962.·6-s + 503.·7-s + 1.44e3·8-s + 5.05e3·9-s − 5.91e3·10-s + 635.·11-s − 5.93·12-s + 3.76e3·13-s − 5.69e3·14-s + 4.45e4·15-s − 1.63e4·16-s − 3.03e4·17-s − 5.72e4·18-s − 3.02e4·19-s − 36.4·20-s + 4.28e4·21-s − 7.18e3·22-s − 9.63e4·23-s + 1.23e5·24-s + 1.95e5·25-s − 4.26e4·26-s + 2.44e5·27-s − 35.0·28-s + ⋯ |
| L(s) = 1 | − 0.999·2-s + 1.82·3-s − 0.000544·4-s + 1.87·5-s − 1.81·6-s + 0.554·7-s + 1.00·8-s + 2.31·9-s − 1.87·10-s + 0.143·11-s − 0.000990·12-s + 0.475·13-s − 0.554·14-s + 3.40·15-s − 0.999·16-s − 1.50·17-s − 2.31·18-s − 1.01·19-s − 0.00101·20-s + 1.00·21-s − 0.143·22-s − 1.65·23-s + 1.82·24-s + 2.50·25-s − 0.475·26-s + 2.38·27-s − 0.000301·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 61 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 61 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(2.823756740\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.823756740\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 61 | \( 1 + 2.26e5T \) |
| good | 2 | \( 1 + 11.3T + 128T^{2} \) |
| 3 | \( 1 - 85.1T + 2.18e3T^{2} \) |
| 5 | \( 1 - 523.T + 7.81e4T^{2} \) |
| 7 | \( 1 - 503.T + 8.23e5T^{2} \) |
| 11 | \( 1 - 635.T + 1.94e7T^{2} \) |
| 13 | \( 1 - 3.76e3T + 6.27e7T^{2} \) |
| 17 | \( 1 + 3.03e4T + 4.10e8T^{2} \) |
| 19 | \( 1 + 3.02e4T + 8.93e8T^{2} \) |
| 23 | \( 1 + 9.63e4T + 3.40e9T^{2} \) |
| 29 | \( 1 - 6.08e4T + 1.72e10T^{2} \) |
| 31 | \( 1 - 1.97e5T + 2.75e10T^{2} \) |
| 37 | \( 1 + 1.12e5T + 9.49e10T^{2} \) |
| 41 | \( 1 + 2.48e5T + 1.94e11T^{2} \) |
| 43 | \( 1 + 4.27e5T + 2.71e11T^{2} \) |
| 47 | \( 1 - 9.61e5T + 5.06e11T^{2} \) |
| 53 | \( 1 + 6.35e5T + 1.17e12T^{2} \) |
| 59 | \( 1 - 2.15e5T + 2.48e12T^{2} \) |
| 67 | \( 1 - 6.17e5T + 6.06e12T^{2} \) |
| 71 | \( 1 - 3.71e6T + 9.09e12T^{2} \) |
| 73 | \( 1 + 3.02e6T + 1.10e13T^{2} \) |
| 79 | \( 1 + 2.98e6T + 1.92e13T^{2} \) |
| 83 | \( 1 + 7.47e4T + 2.71e13T^{2} \) |
| 89 | \( 1 + 1.08e7T + 4.42e13T^{2} \) |
| 97 | \( 1 - 1.56e7T + 8.07e13T^{2} \) |
| 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
−13.76249682512031017182929488667, −13.08674687879280158847908943020, −10.49828694121296789697234049075, −9.757518266322662709138874870646, −8.818651934776523957128599375328, −8.278400065137968045241990375033, −6.60383120723516614673772052409, −4.43096910294411237370849928926, −2.28765560219081351487506586243, −1.59794708228845112525402192455,
1.59794708228845112525402192455, 2.28765560219081351487506586243, 4.43096910294411237370849928926, 6.60383120723516614673772052409, 8.278400065137968045241990375033, 8.818651934776523957128599375328, 9.757518266322662709138874870646, 10.49828694121296789697234049075, 13.08674687879280158847908943020, 13.76249682512031017182929488667
