| L(s) = 1 | − 2-s − 3.06·3-s + 4-s − 0.106·5-s + 3.06·6-s + 0.667·7-s − 8-s + 6.41·9-s + 0.106·10-s − 3.81·11-s − 3.06·12-s − 2.35·13-s − 0.667·14-s + 0.328·15-s + 16-s − 7.95·17-s − 6.41·18-s − 0.927·19-s − 0.106·20-s − 2.04·21-s + 3.81·22-s + 2.46·23-s + 3.06·24-s − 4.98·25-s + 2.35·26-s − 10.4·27-s + 0.667·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1.77·3-s + 0.5·4-s − 0.0478·5-s + 1.25·6-s + 0.252·7-s − 0.353·8-s + 2.13·9-s + 0.0338·10-s − 1.15·11-s − 0.885·12-s − 0.653·13-s − 0.178·14-s + 0.0847·15-s + 0.250·16-s − 1.92·17-s − 1.51·18-s − 0.212·19-s − 0.0239·20-s − 0.446·21-s + 0.814·22-s + 0.513·23-s + 0.626·24-s − 0.997·25-s + 0.461·26-s − 2.01·27-s + 0.126·28-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
| \(L(1)\) |
\(\approx\) |
\(0.1917556912\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1917556912\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + 3.06T + 3T^{2} \) |
| 5 | \( 1 + 0.106T + 5T^{2} \) |
| 7 | \( 1 - 0.667T + 7T^{2} \) |
| 11 | \( 1 + 3.81T + 11T^{2} \) |
| 13 | \( 1 + 2.35T + 13T^{2} \) |
| 17 | \( 1 + 7.95T + 17T^{2} \) |
| 19 | \( 1 + 0.927T + 19T^{2} \) |
| 23 | \( 1 - 2.46T + 23T^{2} \) |
| 29 | \( 1 - 2.73T + 29T^{2} \) |
| 31 | \( 1 - 0.338T + 31T^{2} \) |
| 37 | \( 1 + 3.73T + 37T^{2} \) |
| 41 | \( 1 - 4.79T + 41T^{2} \) |
| 43 | \( 1 + 6.21T + 43T^{2} \) |
| 47 | \( 1 + 6.54T + 47T^{2} \) |
| 53 | \( 1 - 13.6T + 53T^{2} \) |
| 59 | \( 1 + 9.98T + 59T^{2} \) |
| 61 | \( 1 - 11.3T + 61T^{2} \) |
| 67 | \( 1 + 6.58T + 67T^{2} \) |
| 71 | \( 1 - 3.82T + 71T^{2} \) |
| 73 | \( 1 + 12.5T + 73T^{2} \) |
| 79 | \( 1 + 17.3T + 79T^{2} \) |
| 83 | \( 1 - 14.6T + 83T^{2} \) |
| 89 | \( 1 - 8.59T + 89T^{2} \) |
| 97 | \( 1 + 0.720T + 97T^{2} \) |
| 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
−8.389611682647197943342082254899, −7.59275430361512639463738891674, −6.87715284053446127544377640306, −6.37979938254686733855994025771, −5.47376291455776402605834346396, −4.91152760249844224764792159460, −4.18664232139027388083116391779, −2.63289935238157943190313784156, −1.68137355604486091521744595682, −0.29858764804992845229239084503,
0.29858764804992845229239084503, 1.68137355604486091521744595682, 2.63289935238157943190313784156, 4.18664232139027388083116391779, 4.91152760249844224764792159460, 5.47376291455776402605834346396, 6.37979938254686733855994025771, 6.87715284053446127544377640306, 7.59275430361512639463738891674, 8.389611682647197943342082254899
