| L(s) = 1 | − 2-s + 1.23·3-s + 4-s − 1.23·5-s − 1.23·6-s + 0.431·7-s − 8-s − 1.48·9-s + 1.23·10-s − 1.66·11-s + 1.23·12-s − 3.66·13-s − 0.431·14-s − 1.51·15-s + 16-s + 6.60·17-s + 1.48·18-s + 3.71·19-s − 1.23·20-s + 0.531·21-s + 1.66·22-s − 23-s − 1.23·24-s − 3.48·25-s + 3.66·26-s − 5.52·27-s + 0.431·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.711·3-s + 0.5·4-s − 0.551·5-s − 0.503·6-s + 0.163·7-s − 0.353·8-s − 0.493·9-s + 0.389·10-s − 0.501·11-s + 0.355·12-s − 1.01·13-s − 0.115·14-s − 0.392·15-s + 0.250·16-s + 1.60·17-s + 0.349·18-s + 0.851·19-s − 0.275·20-s + 0.115·21-s + 0.354·22-s − 0.208·23-s − 0.251·24-s − 0.696·25-s + 0.718·26-s − 1.06·27-s + 0.0815·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1334 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1334 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 23 | \( 1 + T \) |
| 29 | \( 1 + T \) |
| good | 3 | \( 1 - 1.23T + 3T^{2} \) |
| 5 | \( 1 + 1.23T + 5T^{2} \) |
| 7 | \( 1 - 0.431T + 7T^{2} \) |
| 11 | \( 1 + 1.66T + 11T^{2} \) |
| 13 | \( 1 + 3.66T + 13T^{2} \) |
| 17 | \( 1 - 6.60T + 17T^{2} \) |
| 19 | \( 1 - 3.71T + 19T^{2} \) |
| 31 | \( 1 + 9.66T + 31T^{2} \) |
| 37 | \( 1 + 8.22T + 37T^{2} \) |
| 41 | \( 1 - 1.75T + 41T^{2} \) |
| 43 | \( 1 - 5.76T + 43T^{2} \) |
| 47 | \( 1 + 8.22T + 47T^{2} \) |
| 53 | \( 1 + 0.459T + 53T^{2} \) |
| 59 | \( 1 - 3.32T + 59T^{2} \) |
| 61 | \( 1 + 5.56T + 61T^{2} \) |
| 67 | \( 1 + 8.60T + 67T^{2} \) |
| 71 | \( 1 + 4.96T + 71T^{2} \) |
| 73 | \( 1 + 2.24T + 73T^{2} \) |
| 79 | \( 1 - 0.195T + 79T^{2} \) |
| 83 | \( 1 + 5.42T + 83T^{2} \) |
| 89 | \( 1 - 10.0T + 89T^{2} \) |
| 97 | \( 1 - 4.92T + 97T^{2} \) |
| 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
−9.244719652037232232885181293337, −8.317391238575865423989064331330, −7.60726443409318096219633265887, −7.37612943913926103959057289748, −5.84201936963953773534347235372, −5.11874407967946673073661095953, −3.63403408190170291015972460662, −2.95558543977636762496948119907, −1.77274594105414892106599143038, 0,
1.77274594105414892106599143038, 2.95558543977636762496948119907, 3.63403408190170291015972460662, 5.11874407967946673073661095953, 5.84201936963953773534347235372, 7.37612943913926103959057289748, 7.60726443409318096219633265887, 8.317391238575865423989064331330, 9.244719652037232232885181293337
