| L(s) = 1 | + 2-s + 0.554·3-s + 4-s + 1.35·5-s + 0.554·6-s − 7-s + 8-s − 2.69·9-s + 1.35·10-s + 1.80·11-s + 0.554·12-s − 14-s + 0.753·15-s + 16-s + 5.29·17-s − 2.69·18-s + 1.95·19-s + 1.35·20-s − 0.554·21-s + 1.80·22-s + 4.85·23-s + 0.554·24-s − 3.15·25-s − 3.15·27-s − 28-s + 5.96·29-s + 0.753·30-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.320·3-s + 0.5·4-s + 0.606·5-s + 0.226·6-s − 0.377·7-s + 0.353·8-s − 0.897·9-s + 0.429·10-s + 0.543·11-s + 0.160·12-s − 0.267·14-s + 0.194·15-s + 0.250·16-s + 1.28·17-s − 0.634·18-s + 0.447·19-s + 0.303·20-s − 0.121·21-s + 0.384·22-s + 1.01·23-s + 0.113·24-s − 0.631·25-s − 0.607·27-s − 0.188·28-s + 1.10·29-s + 0.137·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2366 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2366 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.510409636\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.510409636\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 7 | \( 1 + T \) |
| 13 | \( 1 \) |
| good | 3 | \( 1 - 0.554T + 3T^{2} \) |
| 5 | \( 1 - 1.35T + 5T^{2} \) |
| 11 | \( 1 - 1.80T + 11T^{2} \) |
| 17 | \( 1 - 5.29T + 17T^{2} \) |
| 19 | \( 1 - 1.95T + 19T^{2} \) |
| 23 | \( 1 - 4.85T + 23T^{2} \) |
| 29 | \( 1 - 5.96T + 29T^{2} \) |
| 31 | \( 1 + 3.63T + 31T^{2} \) |
| 37 | \( 1 - 7.75T + 37T^{2} \) |
| 41 | \( 1 - 0.0392T + 41T^{2} \) |
| 43 | \( 1 - 2.26T + 43T^{2} \) |
| 47 | \( 1 - 10.5T + 47T^{2} \) |
| 53 | \( 1 + 3.66T + 53T^{2} \) |
| 59 | \( 1 - 4.07T + 59T^{2} \) |
| 61 | \( 1 + 11.5T + 61T^{2} \) |
| 67 | \( 1 + 2.22T + 67T^{2} \) |
| 71 | \( 1 - 7.78T + 71T^{2} \) |
| 73 | \( 1 - 14.1T + 73T^{2} \) |
| 79 | \( 1 + 12.0T + 79T^{2} \) |
| 83 | \( 1 - 2.85T + 83T^{2} \) |
| 89 | \( 1 + 6.61T + 89T^{2} \) |
| 97 | \( 1 + 3.64T + 97T^{2} \) |
| 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
−9.139156178391628246486113408815, −8.122672452016226479949175296911, −7.39434499374166393153848589420, −6.43191263404335210442896837874, −5.79185349235949292139111204141, −5.18429092421811213887281254807, −4.01947128822591181734032450504, −3.16101185412149583870684179121, −2.48481634720681423580776757605, −1.14421363680449441349199471809,
1.14421363680449441349199471809, 2.48481634720681423580776757605, 3.16101185412149583870684179121, 4.01947128822591181734032450504, 5.18429092421811213887281254807, 5.79185349235949292139111204141, 6.43191263404335210442896837874, 7.39434499374166393153848589420, 8.122672452016226479949175296911, 9.139156178391628246486113408815
