| L(s) = 1 | + 6.63·2-s + 19.8·3-s + 12·4-s + 132·6-s − 59.6·7-s − 132.·8-s + 153·9-s + 252·11-s + 238.·12-s + 119.·13-s − 396·14-s − 1.26e3·16-s − 689.·17-s + 1.01e3·18-s + 220·19-s − 1.18e3·21-s + 1.67e3·22-s − 2.43e3·23-s − 2.63e3·24-s + 792·26-s − 1.79e3·27-s − 716.·28-s + 6.93e3·29-s + 6.75e3·31-s − 4.13e3·32-s + 5.01e3·33-s − 4.57e3·34-s + ⋯ |
| L(s) = 1 | + 1.17·2-s + 1.27·3-s + 0.375·4-s + 1.49·6-s − 0.460·7-s − 0.732·8-s + 0.629·9-s + 0.627·11-s + 0.478·12-s + 0.195·13-s − 0.539·14-s − 1.23·16-s − 0.578·17-s + 0.738·18-s + 0.139·19-s − 0.587·21-s + 0.736·22-s − 0.959·23-s − 0.935·24-s + 0.229·26-s − 0.472·27-s − 0.172·28-s + 1.53·29-s + 1.26·31-s − 0.714·32-s + 0.801·33-s − 0.678·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 25 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 25 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(3.100156089\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.100156089\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| good | 2 | \( 1 - 6.63T + 32T^{2} \) |
| 3 | \( 1 - 19.8T + 243T^{2} \) |
| 7 | \( 1 + 59.6T + 1.68e4T^{2} \) |
| 11 | \( 1 - 252T + 1.61e5T^{2} \) |
| 13 | \( 1 - 119.T + 3.71e5T^{2} \) |
| 17 | \( 1 + 689.T + 1.41e6T^{2} \) |
| 19 | \( 1 - 220T + 2.47e6T^{2} \) |
| 23 | \( 1 + 2.43e3T + 6.43e6T^{2} \) |
| 29 | \( 1 - 6.93e3T + 2.05e7T^{2} \) |
| 31 | \( 1 - 6.75e3T + 2.86e7T^{2} \) |
| 37 | \( 1 - 1.39e4T + 6.93e7T^{2} \) |
| 41 | \( 1 + 198T + 1.15e8T^{2} \) |
| 43 | \( 1 - 417.T + 1.47e8T^{2} \) |
| 47 | \( 1 + 1.05e4T + 2.29e8T^{2} \) |
| 53 | \( 1 - 5.82e3T + 4.18e8T^{2} \) |
| 59 | \( 1 - 2.46e4T + 7.14e8T^{2} \) |
| 61 | \( 1 + 5.69e3T + 8.44e8T^{2} \) |
| 67 | \( 1 + 4.36e4T + 1.35e9T^{2} \) |
| 71 | \( 1 - 5.33e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 7.09e4T + 2.07e9T^{2} \) |
| 79 | \( 1 + 5.19e4T + 3.07e9T^{2} \) |
| 83 | \( 1 - 6.18e4T + 3.93e9T^{2} \) |
| 89 | \( 1 - 9.99e3T + 5.58e9T^{2} \) |
| 97 | \( 1 + 1.01e5T + 8.58e9T^{2} \) |
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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
−15.96808847967059356798495608751, −14.86265665399919831665133746072, −13.98243777376596457281212555092, −13.18578320242547854231764223809, −11.83242946666747579857149311419, −9.631438910217922301396207177630, −8.392084251191694009100322335679, −6.35109911935816571141696592337, −4.22617927228470560253376187561, −2.83756532588864993917610921779,
2.83756532588864993917610921779, 4.22617927228470560253376187561, 6.35109911935816571141696592337, 8.392084251191694009100322335679, 9.631438910217922301396207177630, 11.83242946666747579857149311419, 13.18578320242547854231764223809, 13.98243777376596457281212555092, 14.86265665399919831665133746072, 15.96808847967059356798495608751
