| L(s) = 1 | − 1.73·5-s − 7-s + 3.46·11-s − 4·13-s + 6.92·17-s − 7·19-s − 2.00·25-s − 10.3·29-s + 31-s + 1.73·35-s + 2·37-s − 1.73·41-s − 10·43-s − 10.3·47-s − 6·49-s − 6.92·53-s − 5.99·55-s + 12.1·59-s − 4·61-s + 6.92·65-s − 4·67-s + 8.66·71-s + 14·73-s − 3.46·77-s − 10·79-s − 17.3·83-s − 11.9·85-s + ⋯ |
| L(s) = 1 | − 0.774·5-s − 0.377·7-s + 1.04·11-s − 1.10·13-s + 1.68·17-s − 1.60·19-s − 0.400·25-s − 1.92·29-s + 0.179·31-s + 0.292·35-s + 0.328·37-s − 0.270·41-s − 1.52·43-s − 1.51·47-s − 0.857·49-s − 0.951·53-s − 0.809·55-s + 1.57·59-s − 0.512·61-s + 0.859·65-s − 0.488·67-s + 1.02·71-s + 1.63·73-s − 0.394·77-s − 1.12·79-s − 1.90·83-s − 1.30·85-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1116 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1116 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 31 | \( 1 - T \) |
| good | 5 | \( 1 + 1.73T + 5T^{2} \) |
| 7 | \( 1 + T + 7T^{2} \) |
| 11 | \( 1 - 3.46T + 11T^{2} \) |
| 13 | \( 1 + 4T + 13T^{2} \) |
| 17 | \( 1 - 6.92T + 17T^{2} \) |
| 19 | \( 1 + 7T + 19T^{2} \) |
| 23 | \( 1 + 23T^{2} \) |
| 29 | \( 1 + 10.3T + 29T^{2} \) |
| 37 | \( 1 - 2T + 37T^{2} \) |
| 41 | \( 1 + 1.73T + 41T^{2} \) |
| 43 | \( 1 + 10T + 43T^{2} \) |
| 47 | \( 1 + 10.3T + 47T^{2} \) |
| 53 | \( 1 + 6.92T + 53T^{2} \) |
| 59 | \( 1 - 12.1T + 59T^{2} \) |
| 61 | \( 1 + 4T + 61T^{2} \) |
| 67 | \( 1 + 4T + 67T^{2} \) |
| 71 | \( 1 - 8.66T + 71T^{2} \) |
| 73 | \( 1 - 14T + 73T^{2} \) |
| 79 | \( 1 + 10T + 79T^{2} \) |
| 83 | \( 1 + 17.3T + 83T^{2} \) |
| 89 | \( 1 + 10.3T + 89T^{2} \) |
| 97 | \( 1 - 17T + 97T^{2} \) |
| 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
−9.639202411453674887958093847198, −8.498536492518241227840702268058, −7.78884980354284877523041988989, −6.97697152113646979095696764531, −6.13086434095297418055062470041, −5.04258489583109829506374040236, −3.99856011110211439934302920334, −3.29000551453997748342146833007, −1.79701755778497925357944282053, 0,
1.79701755778497925357944282053, 3.29000551453997748342146833007, 3.99856011110211439934302920334, 5.04258489583109829506374040236, 6.13086434095297418055062470041, 6.97697152113646979095696764531, 7.78884980354284877523041988989, 8.498536492518241227840702268058, 9.639202411453674887958093847198
