| L(s) = 1 | + 2-s − 1.38·3-s + 4-s − 1.46·5-s − 1.38·6-s + 0.773·7-s + 8-s − 1.09·9-s − 1.46·10-s + 2.99·11-s − 1.38·12-s + 3.76·13-s + 0.773·14-s + 2.01·15-s + 16-s − 2.55·17-s − 1.09·18-s + 2.03·19-s − 1.46·20-s − 1.06·21-s + 2.99·22-s − 6.63·23-s − 1.38·24-s − 2.86·25-s + 3.76·26-s + 5.65·27-s + 0.773·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.797·3-s + 0.5·4-s − 0.652·5-s − 0.563·6-s + 0.292·7-s + 0.353·8-s − 0.364·9-s − 0.461·10-s + 0.901·11-s − 0.398·12-s + 1.04·13-s + 0.206·14-s + 0.520·15-s + 0.250·16-s − 0.618·17-s − 0.257·18-s + 0.466·19-s − 0.326·20-s − 0.233·21-s + 0.637·22-s − 1.38·23-s − 0.281·24-s − 0.573·25-s + 0.738·26-s + 1.08·27-s + 0.146·28-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
| \(L(1)\) |
\(\approx\) |
\(2.005520353\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.005520353\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + 1.38T + 3T^{2} \) |
| 5 | \( 1 + 1.46T + 5T^{2} \) |
| 7 | \( 1 - 0.773T + 7T^{2} \) |
| 11 | \( 1 - 2.99T + 11T^{2} \) |
| 13 | \( 1 - 3.76T + 13T^{2} \) |
| 17 | \( 1 + 2.55T + 17T^{2} \) |
| 19 | \( 1 - 2.03T + 19T^{2} \) |
| 23 | \( 1 + 6.63T + 23T^{2} \) |
| 29 | \( 1 - 3.84T + 29T^{2} \) |
| 31 | \( 1 + 0.975T + 31T^{2} \) |
| 37 | \( 1 - 10.7T + 37T^{2} \) |
| 41 | \( 1 - 1.33T + 41T^{2} \) |
| 43 | \( 1 + 7.94T + 43T^{2} \) |
| 47 | \( 1 - 5.68T + 47T^{2} \) |
| 53 | \( 1 - 4.50T + 53T^{2} \) |
| 59 | \( 1 + 4.26T + 59T^{2} \) |
| 61 | \( 1 + 4.68T + 61T^{2} \) |
| 67 | \( 1 - 11.2T + 67T^{2} \) |
| 71 | \( 1 - 7.35T + 71T^{2} \) |
| 73 | \( 1 + 2.27T + 73T^{2} \) |
| 79 | \( 1 + 7.15T + 79T^{2} \) |
| 83 | \( 1 - 12.6T + 83T^{2} \) |
| 89 | \( 1 - 10.4T + 89T^{2} \) |
| 97 | \( 1 - 4.58T + 97T^{2} \) |
| 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
−8.261716903772945672928544757143, −7.72377165995270432724119698454, −6.60160372869145775412762949210, −6.23221535004337580421880411671, −5.54132504904300688702432614211, −4.58492550606141950489910557664, −4.01395176130476191764389706243, −3.23000551264679744202218228557, −1.98343071033183284764449476938, −0.76717984814652669462017968622,
0.76717984814652669462017968622, 1.98343071033183284764449476938, 3.23000551264679744202218228557, 4.01395176130476191764389706243, 4.58492550606141950489910557664, 5.54132504904300688702432614211, 6.23221535004337580421880411671, 6.60160372869145775412762949210, 7.72377165995270432724119698454, 8.261716903772945672928544757143
