| L(s) = 1 | − 1.40·2-s + 2.97·3-s − 0.0157·4-s − 5-s − 4.18·6-s + 7-s + 2.83·8-s + 5.83·9-s + 1.40·10-s − 0.0467·12-s + 6.87·13-s − 1.40·14-s − 2.97·15-s − 3.96·16-s + 7.90·17-s − 8.21·18-s − 3.35·19-s + 0.0157·20-s + 2.97·21-s + 2.29·23-s + 8.43·24-s + 25-s − 9.68·26-s + 8.41·27-s − 0.0157·28-s + 4.72·29-s + 4.18·30-s + ⋯ |
| L(s) = 1 | − 0.996·2-s + 1.71·3-s − 0.00786·4-s − 0.447·5-s − 1.70·6-s + 0.377·7-s + 1.00·8-s + 1.94·9-s + 0.445·10-s − 0.0134·12-s + 1.90·13-s − 0.376·14-s − 0.767·15-s − 0.992·16-s + 1.91·17-s − 1.93·18-s − 0.769·19-s + 0.00351·20-s + 0.648·21-s + 0.479·23-s + 1.72·24-s + 0.200·25-s − 1.89·26-s + 1.62·27-s − 0.00297·28-s + 0.878·29-s + 0.764·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4235 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4235 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.375966113\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.375966113\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + T \) |
| 7 | \( 1 - T \) |
| 11 | \( 1 \) |
| good | 2 | \( 1 + 1.40T + 2T^{2} \) |
| 3 | \( 1 - 2.97T + 3T^{2} \) |
| 13 | \( 1 - 6.87T + 13T^{2} \) |
| 17 | \( 1 - 7.90T + 17T^{2} \) |
| 19 | \( 1 + 3.35T + 19T^{2} \) |
| 23 | \( 1 - 2.29T + 23T^{2} \) |
| 29 | \( 1 - 4.72T + 29T^{2} \) |
| 31 | \( 1 - 0.522T + 31T^{2} \) |
| 37 | \( 1 + 6.97T + 37T^{2} \) |
| 41 | \( 1 - 3.27T + 41T^{2} \) |
| 43 | \( 1 + 7.28T + 43T^{2} \) |
| 47 | \( 1 - 3.87T + 47T^{2} \) |
| 53 | \( 1 + 7.36T + 53T^{2} \) |
| 59 | \( 1 - 4.29T + 59T^{2} \) |
| 61 | \( 1 - 6.79T + 61T^{2} \) |
| 67 | \( 1 + 0.907T + 67T^{2} \) |
| 71 | \( 1 + 10.9T + 71T^{2} \) |
| 73 | \( 1 + 2.36T + 73T^{2} \) |
| 79 | \( 1 - 14.8T + 79T^{2} \) |
| 83 | \( 1 + 9.12T + 83T^{2} \) |
| 89 | \( 1 + 5.32T + 89T^{2} \) |
| 97 | \( 1 + 4.21T + 97T^{2} \) |
| 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
−8.406476480579512868426513009944, −8.080269500410073070367333081763, −7.41550520569462978565093715702, −6.59129013032853263605200518679, −5.35212903343321741708237189550, −4.27245715272558751782350212839, −3.67399650480250251109522291388, −2.96852650180932388395407422222, −1.67763621142628704183540647523, −1.06931384510940210284666866534,
1.06931384510940210284666866534, 1.67763621142628704183540647523, 2.96852650180932388395407422222, 3.67399650480250251109522291388, 4.27245715272558751782350212839, 5.35212903343321741708237189550, 6.59129013032853263605200518679, 7.41550520569462978565093715702, 8.080269500410073070367333081763, 8.406476480579512868426513009944
