| L(s) = 1 | + 4·2-s + 16·4-s − 4.15·5-s − 174.·7-s + 64·8-s − 16.6·10-s + 235.·11-s + 1.02e3·13-s − 699.·14-s + 256·16-s − 1.45e3·17-s + 1.04e3·19-s − 66.4·20-s + 941.·22-s + 59.4·23-s − 3.10e3·25-s + 4.11e3·26-s − 2.79e3·28-s − 4.19e3·29-s + 961·31-s + 1.02e3·32-s − 5.82e3·34-s + 726.·35-s + 7.79e3·37-s + 4.19e3·38-s − 265.·40-s + 4.01e3·41-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.5·4-s − 0.0743·5-s − 1.34·7-s + 0.353·8-s − 0.0525·10-s + 0.586·11-s + 1.68·13-s − 0.954·14-s + 0.250·16-s − 1.22·17-s + 0.667·19-s − 0.0371·20-s + 0.414·22-s + 0.0234·23-s − 0.994·25-s + 1.19·26-s − 0.674·28-s − 0.925·29-s + 0.179·31-s + 0.176·32-s − 0.863·34-s + 0.100·35-s + 0.936·37-s + 0.471·38-s − 0.0262·40-s + 0.372·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 558 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 558 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(3.072177516\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.072177516\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - 4T \) |
| 3 | \( 1 \) |
| 31 | \( 1 - 961T \) |
| good | 5 | \( 1 + 4.15T + 3.12e3T^{2} \) |
| 7 | \( 1 + 174.T + 1.68e4T^{2} \) |
| 11 | \( 1 - 235.T + 1.61e5T^{2} \) |
| 13 | \( 1 - 1.02e3T + 3.71e5T^{2} \) |
| 17 | \( 1 + 1.45e3T + 1.41e6T^{2} \) |
| 19 | \( 1 - 1.04e3T + 2.47e6T^{2} \) |
| 23 | \( 1 - 59.4T + 6.43e6T^{2} \) |
| 29 | \( 1 + 4.19e3T + 2.05e7T^{2} \) |
| 37 | \( 1 - 7.79e3T + 6.93e7T^{2} \) |
| 41 | \( 1 - 4.01e3T + 1.15e8T^{2} \) |
| 43 | \( 1 + 1.99e3T + 1.47e8T^{2} \) |
| 47 | \( 1 + 1.33e4T + 2.29e8T^{2} \) |
| 53 | \( 1 - 3.18e4T + 4.18e8T^{2} \) |
| 59 | \( 1 - 3.54e4T + 7.14e8T^{2} \) |
| 61 | \( 1 - 4.58e4T + 8.44e8T^{2} \) |
| 67 | \( 1 - 5.55e4T + 1.35e9T^{2} \) |
| 71 | \( 1 - 4.70e4T + 1.80e9T^{2} \) |
| 73 | \( 1 - 7.92e4T + 2.07e9T^{2} \) |
| 79 | \( 1 - 814.T + 3.07e9T^{2} \) |
| 83 | \( 1 + 4.76e4T + 3.93e9T^{2} \) |
| 89 | \( 1 - 4.60e4T + 5.58e9T^{2} \) |
| 97 | \( 1 - 5.42e4T + 8.58e9T^{2} \) |
| 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
−9.952422203527908242539780642614, −9.208419805629896295642314695887, −8.211440817056644894688558393153, −6.88858756405555017954231642933, −6.35360982806407237004521214339, −5.50868593010594948028450599155, −3.97736103170545923293736087580, −3.56064198531007901750649619636, −2.23234634629462648781433688589, −0.77411588563590381436030876095,
0.77411588563590381436030876095, 2.23234634629462648781433688589, 3.56064198531007901750649619636, 3.97736103170545923293736087580, 5.50868593010594948028450599155, 6.35360982806407237004521214339, 6.88858756405555017954231642933, 8.211440817056644894688558393153, 9.208419805629896295642314695887, 9.952422203527908242539780642614
