L(s) = 1 | + 8·2-s − 27·3-s + 64·4-s + 125·5-s − 216·6-s − 1.16e3·7-s + 512·8-s + 729·9-s + 1.00e3·10-s − 144.·11-s − 1.72e3·12-s − 3.04e3·13-s − 9.32e3·14-s − 3.37e3·15-s + 4.09e3·16-s + 6.70e3·17-s + 5.83e3·18-s + 6.85e3·19-s + 8.00e3·20-s + 3.14e4·21-s − 1.15e3·22-s + 1.54e4·23-s − 1.38e4·24-s + 1.56e4·25-s − 2.43e4·26-s − 1.96e4·27-s − 7.46e4·28-s + ⋯ |
L(s) = 1 | + 0.707·2-s − 0.577·3-s + 0.5·4-s + 0.447·5-s − 0.408·6-s − 1.28·7-s + 0.353·8-s + 0.333·9-s + 0.316·10-s − 0.0326·11-s − 0.288·12-s − 0.384·13-s − 0.908·14-s − 0.258·15-s + 0.250·16-s + 0.330·17-s + 0.235·18-s + 0.229·19-s + 0.223·20-s + 0.741·21-s − 0.0230·22-s + 0.264·23-s − 0.204·24-s + 0.199·25-s − 0.272·26-s − 0.192·27-s − 0.642·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(4)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{9}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - 8T \) |
| 3 | \( 1 + 27T \) |
| 5 | \( 1 - 125T \) |
| 19 | \( 1 - 6.85e3T \) |
good | 7 | \( 1 + 1.16e3T + 8.23e5T^{2} \) |
| 11 | \( 1 + 144.T + 1.94e7T^{2} \) |
| 13 | \( 1 + 3.04e3T + 6.27e7T^{2} \) |
| 17 | \( 1 - 6.70e3T + 4.10e8T^{2} \) |
| 23 | \( 1 - 1.54e4T + 3.40e9T^{2} \) |
| 29 | \( 1 - 1.80e5T + 1.72e10T^{2} \) |
| 31 | \( 1 - 1.73e5T + 2.75e10T^{2} \) |
| 37 | \( 1 + 4.52e5T + 9.49e10T^{2} \) |
| 41 | \( 1 - 3.38e5T + 1.94e11T^{2} \) |
| 43 | \( 1 + 3.57e5T + 2.71e11T^{2} \) |
| 47 | \( 1 + 4.86e5T + 5.06e11T^{2} \) |
| 53 | \( 1 + 1.80e6T + 1.17e12T^{2} \) |
| 59 | \( 1 - 1.20e6T + 2.48e12T^{2} \) |
| 61 | \( 1 + 6.14e5T + 3.14e12T^{2} \) |
| 67 | \( 1 - 2.28e6T + 6.06e12T^{2} \) |
| 71 | \( 1 - 2.30e6T + 9.09e12T^{2} \) |
| 73 | \( 1 + 1.41e6T + 1.10e13T^{2} \) |
| 79 | \( 1 + 5.82e6T + 1.92e13T^{2} \) |
| 83 | \( 1 - 5.65e6T + 2.71e13T^{2} \) |
| 89 | \( 1 - 5.01e6T + 4.42e13T^{2} \) |
| 97 | \( 1 + 5.52e6T + 8.07e13T^{2} \) |
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
−9.510285057239034544370821287062, −8.181786172098699667234117936981, −6.87424415374293487561554112058, −6.46226604641903186924078677140, −5.50362967236820008764742892772, −4.66802448040047593795174210712, −3.43757940186406572271588510180, −2.61672442849577308560504444531, −1.22754601498258425998302544240, 0,
1.22754601498258425998302544240, 2.61672442849577308560504444531, 3.43757940186406572271588510180, 4.66802448040047593795174210712, 5.50362967236820008764742892772, 6.46226604641903186924078677140, 6.87424415374293487561554112058, 8.181786172098699667234117936981, 9.510285057239034544370821287062