| L(s) = 1 | + 1.96·3-s + 0.807·5-s + 3.34·7-s + 0.848·9-s + 4.04·11-s − 4.44·13-s + 1.58·15-s − 5.91·17-s + 2.98·19-s + 6.55·21-s + 7.45·23-s − 4.34·25-s − 4.22·27-s + 7.98·29-s + 9.58·31-s + 7.93·33-s + 2.69·35-s + 2.67·37-s − 8.71·39-s + 3.85·41-s + 9.57·43-s + 0.685·45-s − 9.46·47-s + 4.17·49-s − 11.6·51-s − 9.89·53-s + 3.26·55-s + ⋯ |
| L(s) = 1 | + 1.13·3-s + 0.361·5-s + 1.26·7-s + 0.282·9-s + 1.21·11-s − 1.23·13-s + 0.408·15-s − 1.43·17-s + 0.684·19-s + 1.43·21-s + 1.55·23-s − 0.869·25-s − 0.812·27-s + 1.48·29-s + 1.72·31-s + 1.38·33-s + 0.456·35-s + 0.440·37-s − 1.39·39-s + 0.601·41-s + 1.46·43-s + 0.102·45-s − 1.38·47-s + 0.596·49-s − 1.62·51-s − 1.35·53-s + 0.440·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2008 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.263116508\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.263116508\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 251 | \( 1 + T \) |
| good | 3 | \( 1 - 1.96T + 3T^{2} \) |
| 5 | \( 1 - 0.807T + 5T^{2} \) |
| 7 | \( 1 - 3.34T + 7T^{2} \) |
| 11 | \( 1 - 4.04T + 11T^{2} \) |
| 13 | \( 1 + 4.44T + 13T^{2} \) |
| 17 | \( 1 + 5.91T + 17T^{2} \) |
| 19 | \( 1 - 2.98T + 19T^{2} \) |
| 23 | \( 1 - 7.45T + 23T^{2} \) |
| 29 | \( 1 - 7.98T + 29T^{2} \) |
| 31 | \( 1 - 9.58T + 31T^{2} \) |
| 37 | \( 1 - 2.67T + 37T^{2} \) |
| 41 | \( 1 - 3.85T + 41T^{2} \) |
| 43 | \( 1 - 9.57T + 43T^{2} \) |
| 47 | \( 1 + 9.46T + 47T^{2} \) |
| 53 | \( 1 + 9.89T + 53T^{2} \) |
| 59 | \( 1 - 6.88T + 59T^{2} \) |
| 61 | \( 1 + 7.56T + 61T^{2} \) |
| 67 | \( 1 + 8.97T + 67T^{2} \) |
| 71 | \( 1 + 14.1T + 71T^{2} \) |
| 73 | \( 1 - 3.51T + 73T^{2} \) |
| 79 | \( 1 + 9.64T + 79T^{2} \) |
| 83 | \( 1 - 9.42T + 83T^{2} \) |
| 89 | \( 1 - 0.414T + 89T^{2} \) |
| 97 | \( 1 - 8.39T + 97T^{2} \) |
| 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
−9.165017840849958547495078668293, −8.418471435110756162171091767057, −7.76058328462529451439117474427, −6.96835634394070620124005383601, −6.07277462553899808715320693484, −4.74853287830287171222053832696, −4.44572364973434917233936086158, −3.03241213708768172998373554837, −2.32460023902098225477319524273, −1.29267055865463392839493885330,
1.29267055865463392839493885330, 2.32460023902098225477319524273, 3.03241213708768172998373554837, 4.44572364973434917233936086158, 4.74853287830287171222053832696, 6.07277462553899808715320693484, 6.96835634394070620124005383601, 7.76058328462529451439117474427, 8.418471435110756162171091767057, 9.165017840849958547495078668293
