| L(s) = 1 | + 0.642·2-s + 1.06·3-s − 1.58·4-s − 0.849·5-s + 0.684·6-s − 0.754·7-s − 2.30·8-s − 1.86·9-s − 0.546·10-s − 5.48·11-s − 1.68·12-s + 6.40·13-s − 0.485·14-s − 0.904·15-s + 1.69·16-s − 5.81·17-s − 1.19·18-s − 1.45·19-s + 1.34·20-s − 0.803·21-s − 3.52·22-s − 5.24·23-s − 2.45·24-s − 4.27·25-s + 4.11·26-s − 5.18·27-s + 1.19·28-s + ⋯ |
| L(s) = 1 | + 0.454·2-s + 0.614·3-s − 0.793·4-s − 0.379·5-s + 0.279·6-s − 0.285·7-s − 0.815·8-s − 0.621·9-s − 0.172·10-s − 1.65·11-s − 0.487·12-s + 1.77·13-s − 0.129·14-s − 0.233·15-s + 0.422·16-s − 1.41·17-s − 0.282·18-s − 0.332·19-s + 0.301·20-s − 0.175·21-s − 0.751·22-s − 1.09·23-s − 0.501·24-s − 0.855·25-s + 0.807·26-s − 0.997·27-s + 0.226·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 619 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 619 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 619 | \( 1 + T \) |
| good | 2 | \( 1 - 0.642T + 2T^{2} \) |
| 3 | \( 1 - 1.06T + 3T^{2} \) |
| 5 | \( 1 + 0.849T + 5T^{2} \) |
| 7 | \( 1 + 0.754T + 7T^{2} \) |
| 11 | \( 1 + 5.48T + 11T^{2} \) |
| 13 | \( 1 - 6.40T + 13T^{2} \) |
| 17 | \( 1 + 5.81T + 17T^{2} \) |
| 19 | \( 1 + 1.45T + 19T^{2} \) |
| 23 | \( 1 + 5.24T + 23T^{2} \) |
| 29 | \( 1 + 2.02T + 29T^{2} \) |
| 31 | \( 1 - 8.10T + 31T^{2} \) |
| 37 | \( 1 - 2.11T + 37T^{2} \) |
| 41 | \( 1 + 11.5T + 41T^{2} \) |
| 43 | \( 1 - 11.1T + 43T^{2} \) |
| 47 | \( 1 - 3.15T + 47T^{2} \) |
| 53 | \( 1 - 3.12T + 53T^{2} \) |
| 59 | \( 1 - 0.334T + 59T^{2} \) |
| 61 | \( 1 - 8.07T + 61T^{2} \) |
| 67 | \( 1 + 10.5T + 67T^{2} \) |
| 71 | \( 1 + 2.31T + 71T^{2} \) |
| 73 | \( 1 + 5.54T + 73T^{2} \) |
| 79 | \( 1 - 14.4T + 79T^{2} \) |
| 83 | \( 1 - 0.466T + 83T^{2} \) |
| 89 | \( 1 - 6.92T + 89T^{2} \) |
| 97 | \( 1 + 13.2T + 97T^{2} \) |
| 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
−10.18053669942377246556868195450, −9.096161438500587940608746853378, −8.361907255354157488530401141598, −7.960615315317751684698115487921, −6.31920660691703687016991667375, −5.58584675281980784051535555555, −4.35727536818589989541204333838, −3.54425137059898419999749466448, −2.47438698761848823415575214515, 0,
2.47438698761848823415575214515, 3.54425137059898419999749466448, 4.35727536818589989541204333838, 5.58584675281980784051535555555, 6.31920660691703687016991667375, 7.960615315317751684698115487921, 8.361907255354157488530401141598, 9.096161438500587940608746853378, 10.18053669942377246556868195450
