| L(s) = 1 | + 2-s + 4-s + 1.73·5-s + 4.73·7-s + 8-s + 1.73·10-s − 4.73·11-s + 4.73·14-s + 16-s + 5.19·17-s + 1.26·19-s + 1.73·20-s − 4.73·22-s − 2.19·23-s − 2.00·25-s + 4.73·28-s + 3·29-s + 2.53·31-s + 32-s + 5.19·34-s + 8.19·35-s + 3·37-s + 1.26·38-s + 1.73·40-s + 0.464·41-s + 6.19·43-s − 4.73·44-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.5·4-s + 0.774·5-s + 1.78·7-s + 0.353·8-s + 0.547·10-s − 1.42·11-s + 1.26·14-s + 0.250·16-s + 1.26·17-s + 0.290·19-s + 0.387·20-s − 1.00·22-s − 0.457·23-s − 0.400·25-s + 0.894·28-s + 0.557·29-s + 0.455·31-s + 0.176·32-s + 0.891·34-s + 1.38·35-s + 0.493·37-s + 0.205·38-s + 0.273·40-s + 0.0724·41-s + 0.944·43-s − 0.713·44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3042 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3042 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.134206033\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.134206033\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 3 | \( 1 \) |
| 13 | \( 1 \) |
| good | 5 | \( 1 - 1.73T + 5T^{2} \) |
| 7 | \( 1 - 4.73T + 7T^{2} \) |
| 11 | \( 1 + 4.73T + 11T^{2} \) |
| 17 | \( 1 - 5.19T + 17T^{2} \) |
| 19 | \( 1 - 1.26T + 19T^{2} \) |
| 23 | \( 1 + 2.19T + 23T^{2} \) |
| 29 | \( 1 - 3T + 29T^{2} \) |
| 31 | \( 1 - 2.53T + 31T^{2} \) |
| 37 | \( 1 - 3T + 37T^{2} \) |
| 41 | \( 1 - 0.464T + 41T^{2} \) |
| 43 | \( 1 - 6.19T + 43T^{2} \) |
| 47 | \( 1 - 1.26T + 47T^{2} \) |
| 53 | \( 1 + 3T + 53T^{2} \) |
| 59 | \( 1 + 13.8T + 59T^{2} \) |
| 61 | \( 1 - 4.80T + 61T^{2} \) |
| 67 | \( 1 - 10.7T + 67T^{2} \) |
| 71 | \( 1 + 8.19T + 71T^{2} \) |
| 73 | \( 1 + 12.1T + 73T^{2} \) |
| 79 | \( 1 + 12.3T + 79T^{2} \) |
| 83 | \( 1 - 11.6T + 83T^{2} \) |
| 89 | \( 1 + 2.53T + 89T^{2} \) |
| 97 | \( 1 - 6T + 97T^{2} \) |
| 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
−8.457923756837291590198052217271, −7.81167041435696055564893916175, −7.44923674779790931281839528546, −6.09263770622711338473562121105, −5.52464120770402188106532231200, −4.97915391130726567710947077228, −4.23707681257375981602545169463, −2.95166451406264835295467339684, −2.16298875804859904915391716491, −1.24825277263668092873573165802,
1.24825277263668092873573165802, 2.16298875804859904915391716491, 2.95166451406264835295467339684, 4.23707681257375981602545169463, 4.97915391130726567710947077228, 5.52464120770402188106532231200, 6.09263770622711338473562121105, 7.44923674779790931281839528546, 7.81167041435696055564893916175, 8.457923756837291590198052217271
