| L(s) = 1 | + 8.33·2-s − 27·3-s − 58.4·4-s + 308.·5-s − 225.·6-s − 968.·7-s − 1.55e3·8-s + 729·9-s + 2.57e3·10-s − 7.95e3·11-s + 1.57e3·12-s − 4.21e3·13-s − 8.07e3·14-s − 8.32e3·15-s − 5.47e3·16-s − 1.95e4·17-s + 6.07e3·18-s − 2.31e4·19-s − 1.80e4·20-s + 2.61e4·21-s − 6.63e4·22-s − 4.99e4·23-s + 4.19e4·24-s + 1.69e4·25-s − 3.51e4·26-s − 1.96e4·27-s + 5.66e4·28-s + ⋯ |
| L(s) = 1 | + 0.736·2-s − 0.577·3-s − 0.456·4-s + 1.10·5-s − 0.425·6-s − 1.06·7-s − 1.07·8-s + 0.333·9-s + 0.812·10-s − 1.80·11-s + 0.263·12-s − 0.531·13-s − 0.786·14-s − 0.636·15-s − 0.334·16-s − 0.963·17-s + 0.245·18-s − 0.774·19-s − 0.503·20-s + 0.616·21-s − 1.32·22-s − 0.856·23-s + 0.619·24-s + 0.216·25-s − 0.391·26-s − 0.192·27-s + 0.487·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 471 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 471 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(0.4512078149\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4512078149\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + 27T \) |
| 157 | \( 1 + 3.86e6T \) |
| good | 2 | \( 1 - 8.33T + 128T^{2} \) |
| 5 | \( 1 - 308.T + 7.81e4T^{2} \) |
| 7 | \( 1 + 968.T + 8.23e5T^{2} \) |
| 11 | \( 1 + 7.95e3T + 1.94e7T^{2} \) |
| 13 | \( 1 + 4.21e3T + 6.27e7T^{2} \) |
| 17 | \( 1 + 1.95e4T + 4.10e8T^{2} \) |
| 19 | \( 1 + 2.31e4T + 8.93e8T^{2} \) |
| 23 | \( 1 + 4.99e4T + 3.40e9T^{2} \) |
| 29 | \( 1 - 1.43e5T + 1.72e10T^{2} \) |
| 31 | \( 1 + 4.64e4T + 2.75e10T^{2} \) |
| 37 | \( 1 - 1.68e5T + 9.49e10T^{2} \) |
| 41 | \( 1 - 2.66e5T + 1.94e11T^{2} \) |
| 43 | \( 1 + 7.52e5T + 2.71e11T^{2} \) |
| 47 | \( 1 + 4.37e5T + 5.06e11T^{2} \) |
| 53 | \( 1 + 6.53e3T + 1.17e12T^{2} \) |
| 59 | \( 1 + 1.08e6T + 2.48e12T^{2} \) |
| 61 | \( 1 - 1.05e6T + 3.14e12T^{2} \) |
| 67 | \( 1 - 4.45e5T + 6.06e12T^{2} \) |
| 71 | \( 1 + 1.61e6T + 9.09e12T^{2} \) |
| 73 | \( 1 + 4.53e6T + 1.10e13T^{2} \) |
| 79 | \( 1 + 4.70e6T + 1.92e13T^{2} \) |
| 83 | \( 1 - 9.28e6T + 2.71e13T^{2} \) |
| 89 | \( 1 - 5.35e6T + 4.42e13T^{2} \) |
| 97 | \( 1 - 1.12e7T + 8.07e13T^{2} \) |
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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
−10.02627827076755149265085213064, −9.210200745823138271884587129815, −8.069886445801894280444057441125, −6.62180132006833552899077198423, −6.00808788338998420405818840527, −5.21016713045627133534831744275, −4.40945248301642145913256120135, −3.00578914827311227118145585963, −2.17069575001655440046394321780, −0.25492916434097252790695365317,
0.25492916434097252790695365317, 2.17069575001655440046394321780, 3.00578914827311227118145585963, 4.40945248301642145913256120135, 5.21016713045627133534831744275, 6.00808788338998420405818840527, 6.62180132006833552899077198423, 8.069886445801894280444057441125, 9.210200745823138271884587129815, 10.02627827076755149265085213064
