L(s) = 1 | − 19.6·2-s + 258.·4-s + 436.·5-s + 956.·7-s − 2.56e3·8-s − 8.58e3·10-s − 1.97e3·11-s − 5.30e3·13-s − 1.88e4·14-s + 1.72e4·16-s + 1.91e4·17-s − 2.14e4·19-s + 1.12e5·20-s + 3.87e4·22-s + 3.43e4·23-s + 1.12e5·25-s + 1.04e5·26-s + 2.47e5·28-s − 4.00e4·29-s − 9.54e4·31-s − 1.20e4·32-s − 3.77e5·34-s + 4.17e5·35-s − 2.69e5·37-s + 4.21e5·38-s − 1.11e6·40-s + 3.60e5·41-s + ⋯ |
L(s) = 1 | − 1.73·2-s + 2.01·4-s + 1.56·5-s + 1.05·7-s − 1.76·8-s − 2.71·10-s − 0.446·11-s − 0.670·13-s − 1.83·14-s + 1.05·16-s + 0.947·17-s − 0.717·19-s + 3.15·20-s + 0.776·22-s + 0.588·23-s + 1.44·25-s + 1.16·26-s + 2.12·28-s − 0.304·29-s − 0.575·31-s − 0.0647·32-s − 1.64·34-s + 1.64·35-s − 0.873·37-s + 1.24·38-s − 2.76·40-s + 0.816·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 531 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 531 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 \) |
| 59 | \( 1 + 2.05e5T \) |
good | 2 | \( 1 + 19.6T + 128T^{2} \) |
| 5 | \( 1 - 436.T + 7.81e4T^{2} \) |
| 7 | \( 1 - 956.T + 8.23e5T^{2} \) |
| 11 | \( 1 + 1.97e3T + 1.94e7T^{2} \) |
| 13 | \( 1 + 5.30e3T + 6.27e7T^{2} \) |
| 17 | \( 1 - 1.91e4T + 4.10e8T^{2} \) |
| 19 | \( 1 + 2.14e4T + 8.93e8T^{2} \) |
| 23 | \( 1 - 3.43e4T + 3.40e9T^{2} \) |
| 29 | \( 1 + 4.00e4T + 1.72e10T^{2} \) |
| 31 | \( 1 + 9.54e4T + 2.75e10T^{2} \) |
| 37 | \( 1 + 2.69e5T + 9.49e10T^{2} \) |
| 41 | \( 1 - 3.60e5T + 1.94e11T^{2} \) |
| 43 | \( 1 + 9.29e5T + 2.71e11T^{2} \) |
| 47 | \( 1 + 5.40e4T + 5.06e11T^{2} \) |
| 53 | \( 1 + 6.80e5T + 1.17e12T^{2} \) |
| 61 | \( 1 + 1.27e6T + 3.14e12T^{2} \) |
| 67 | \( 1 - 6.30e5T + 6.06e12T^{2} \) |
| 71 | \( 1 + 2.05e6T + 9.09e12T^{2} \) |
| 73 | \( 1 + 5.36e6T + 1.10e13T^{2} \) |
| 79 | \( 1 - 4.57e6T + 1.92e13T^{2} \) |
| 83 | \( 1 + 9.49e6T + 2.71e13T^{2} \) |
| 89 | \( 1 - 2.45e6T + 4.42e13T^{2} \) |
| 97 | \( 1 + 1.41e7T + 8.07e13T^{2} \) |
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
−9.318562735172092708029798165633, −8.525990251244530057256478559619, −7.70923160573212043709722887436, −6.84563333487282297356757364960, −5.77414495256800856556121886620, −4.90948809355291715127213268689, −2.81648164372211644347165496125, −1.85192107895209792244282561895, −1.36755121563122431001990874951, 0,
1.36755121563122431001990874951, 1.85192107895209792244282561895, 2.81648164372211644347165496125, 4.90948809355291715127213268689, 5.77414495256800856556121886620, 6.84563333487282297356757364960, 7.70923160573212043709722887436, 8.525990251244530057256478559619, 9.318562735172092708029798165633