| L(s) = 1 | − 2-s + 0.717·3-s + 4-s + 1.07·5-s − 0.717·6-s + 0.484·7-s − 8-s − 2.48·9-s − 1.07·10-s − 1.94·11-s + 0.717·12-s + 0.948·13-s − 0.484·14-s + 0.775·15-s + 16-s + 5.21·17-s + 2.48·18-s + 1.25·19-s + 1.07·20-s + 0.347·21-s + 1.94·22-s + 8.71·23-s − 0.717·24-s − 3.83·25-s − 0.948·26-s − 3.93·27-s + 0.484·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.414·3-s + 0.5·4-s + 0.482·5-s − 0.293·6-s + 0.182·7-s − 0.353·8-s − 0.828·9-s − 0.341·10-s − 0.585·11-s + 0.207·12-s + 0.262·13-s − 0.129·14-s + 0.200·15-s + 0.250·16-s + 1.26·17-s + 0.585·18-s + 0.288·19-s + 0.241·20-s + 0.0758·21-s + 0.414·22-s + 1.81·23-s − 0.146·24-s − 0.766·25-s − 0.185·26-s − 0.757·27-s + 0.0914·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4006 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4006 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.668097791\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.668097791\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 2003 | \( 1 - T \) |
| good | 3 | \( 1 - 0.717T + 3T^{2} \) |
| 5 | \( 1 - 1.07T + 5T^{2} \) |
| 7 | \( 1 - 0.484T + 7T^{2} \) |
| 11 | \( 1 + 1.94T + 11T^{2} \) |
| 13 | \( 1 - 0.948T + 13T^{2} \) |
| 17 | \( 1 - 5.21T + 17T^{2} \) |
| 19 | \( 1 - 1.25T + 19T^{2} \) |
| 23 | \( 1 - 8.71T + 23T^{2} \) |
| 29 | \( 1 + 4.80T + 29T^{2} \) |
| 31 | \( 1 - 5.23T + 31T^{2} \) |
| 37 | \( 1 + 2.14T + 37T^{2} \) |
| 41 | \( 1 - 9.62T + 41T^{2} \) |
| 43 | \( 1 - 5.70T + 43T^{2} \) |
| 47 | \( 1 + 4.44T + 47T^{2} \) |
| 53 | \( 1 + 7.60T + 53T^{2} \) |
| 59 | \( 1 + 7.29T + 59T^{2} \) |
| 61 | \( 1 - 4.42T + 61T^{2} \) |
| 67 | \( 1 - 1.61T + 67T^{2} \) |
| 71 | \( 1 - 3.03T + 71T^{2} \) |
| 73 | \( 1 - 13.9T + 73T^{2} \) |
| 79 | \( 1 + 0.679T + 79T^{2} \) |
| 83 | \( 1 - 5.60T + 83T^{2} \) |
| 89 | \( 1 + 10.1T + 89T^{2} \) |
| 97 | \( 1 + 5.55T + 97T^{2} \) |
| 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
−8.364957252600273823915859681657, −7.88522699100310030387701916196, −7.25695999405900944240344202346, −6.19246916497601818128562840698, −5.62215380491204255685038758353, −4.86369045492782878517893373999, −3.45935363458467501847872313043, −2.88228481296676293639340562134, −1.93977064370421920471986211059, −0.811656030350167816075801352918,
0.811656030350167816075801352918, 1.93977064370421920471986211059, 2.88228481296676293639340562134, 3.45935363458467501847872313043, 4.86369045492782878517893373999, 5.62215380491204255685038758353, 6.19246916497601818128562840698, 7.25695999405900944240344202346, 7.88522699100310030387701916196, 8.364957252600273823915859681657
