| L(s) = 1 | + 2.50·2-s + 0.136·3-s + 4.29·4-s + 0.342·6-s + 4.91·7-s + 5.76·8-s − 2.98·9-s + 3.67·11-s + 0.586·12-s + 2.24·13-s + 12.3·14-s + 5.88·16-s − 7.15·17-s − 7.48·18-s − 3.70·19-s + 0.669·21-s + 9.21·22-s + 0.647·23-s + 0.786·24-s + 5.63·26-s − 0.815·27-s + 21.1·28-s + 7.87·29-s + 3.74·31-s + 3.22·32-s + 0.501·33-s − 17.9·34-s + ⋯ |
| L(s) = 1 | + 1.77·2-s + 0.0787·3-s + 2.14·4-s + 0.139·6-s + 1.85·7-s + 2.03·8-s − 0.993·9-s + 1.10·11-s + 0.169·12-s + 0.622·13-s + 3.29·14-s + 1.47·16-s − 1.73·17-s − 1.76·18-s − 0.849·19-s + 0.146·21-s + 1.96·22-s + 0.134·23-s + 0.160·24-s + 1.10·26-s − 0.157·27-s + 3.99·28-s + 1.46·29-s + 0.672·31-s + 0.569·32-s + 0.0872·33-s − 3.07·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6025 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6025 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(8.039351075\) |
| \(L(\frac12)\) |
\(\approx\) |
\(8.039351075\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| 241 | \( 1 + T \) |
| good | 2 | \( 1 - 2.50T + 2T^{2} \) |
| 3 | \( 1 - 0.136T + 3T^{2} \) |
| 7 | \( 1 - 4.91T + 7T^{2} \) |
| 11 | \( 1 - 3.67T + 11T^{2} \) |
| 13 | \( 1 - 2.24T + 13T^{2} \) |
| 17 | \( 1 + 7.15T + 17T^{2} \) |
| 19 | \( 1 + 3.70T + 19T^{2} \) |
| 23 | \( 1 - 0.647T + 23T^{2} \) |
| 29 | \( 1 - 7.87T + 29T^{2} \) |
| 31 | \( 1 - 3.74T + 31T^{2} \) |
| 37 | \( 1 - 10.1T + 37T^{2} \) |
| 41 | \( 1 + 4.37T + 41T^{2} \) |
| 43 | \( 1 - 6.82T + 43T^{2} \) |
| 47 | \( 1 - 10.8T + 47T^{2} \) |
| 53 | \( 1 + 6.63T + 53T^{2} \) |
| 59 | \( 1 + 10.8T + 59T^{2} \) |
| 61 | \( 1 + 5.33T + 61T^{2} \) |
| 67 | \( 1 + 0.479T + 67T^{2} \) |
| 71 | \( 1 - 14.9T + 71T^{2} \) |
| 73 | \( 1 + 14.0T + 73T^{2} \) |
| 79 | \( 1 - 5.30T + 79T^{2} \) |
| 83 | \( 1 + 13.7T + 83T^{2} \) |
| 89 | \( 1 + 0.507T + 89T^{2} \) |
| 97 | \( 1 + 9.41T + 97T^{2} \) |
| 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
−8.051969547146173865152669414476, −7.07521790551615274502148394189, −6.28394612061826039049285113243, −5.96539342958327409519226810311, −4.94294353306687126912897984990, −4.39681041116399383697014157764, −4.07924750190582683919574565943, −2.81672843121983514152486095273, −2.22514805861715536526560214219, −1.27081634317705660033752620230,
1.27081634317705660033752620230, 2.22514805861715536526560214219, 2.81672843121983514152486095273, 4.07924750190582683919574565943, 4.39681041116399383697014157764, 4.94294353306687126912897984990, 5.96539342958327409519226810311, 6.28394612061826039049285113243, 7.07521790551615274502148394189, 8.051969547146173865152669414476
