| L(s) = 1 | − 6.52·2-s − 27.2·3-s + 10.5·4-s − 25·5-s + 177.·6-s − 162.·7-s + 139.·8-s + 497.·9-s + 163.·10-s − 121·11-s − 287.·12-s + 610.·13-s + 1.06e3·14-s + 680.·15-s − 1.25e3·16-s + 746.·17-s − 3.24e3·18-s − 361·19-s − 264.·20-s + 4.43e3·21-s + 789.·22-s − 4.82e3·23-s − 3.80e3·24-s + 625·25-s − 3.98e3·26-s − 6.91e3·27-s − 1.72e3·28-s + ⋯ |
| L(s) = 1 | − 1.15·2-s − 1.74·3-s + 0.330·4-s − 0.447·5-s + 2.01·6-s − 1.25·7-s + 0.772·8-s + 2.04·9-s + 0.515·10-s − 0.301·11-s − 0.576·12-s + 1.00·13-s + 1.44·14-s + 0.780·15-s − 1.22·16-s + 0.626·17-s − 2.35·18-s − 0.229·19-s − 0.147·20-s + 2.19·21-s + 0.347·22-s − 1.90·23-s − 1.34·24-s + 0.200·25-s − 1.15·26-s − 1.82·27-s − 0.414·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + 25T \) |
| 11 | \( 1 + 121T \) |
| 19 | \( 1 + 361T \) |
| good | 2 | \( 1 + 6.52T + 32T^{2} \) |
| 3 | \( 1 + 27.2T + 243T^{2} \) |
| 7 | \( 1 + 162.T + 1.68e4T^{2} \) |
| 13 | \( 1 - 610.T + 3.71e5T^{2} \) |
| 17 | \( 1 - 746.T + 1.41e6T^{2} \) |
| 23 | \( 1 + 4.82e3T + 6.43e6T^{2} \) |
| 29 | \( 1 + 2.59e3T + 2.05e7T^{2} \) |
| 31 | \( 1 + 8.91e3T + 2.86e7T^{2} \) |
| 37 | \( 1 - 8.67e3T + 6.93e7T^{2} \) |
| 41 | \( 1 - 85.8T + 1.15e8T^{2} \) |
| 43 | \( 1 - 7.57e3T + 1.47e8T^{2} \) |
| 47 | \( 1 + 2.05e4T + 2.29e8T^{2} \) |
| 53 | \( 1 - 2.58e4T + 4.18e8T^{2} \) |
| 59 | \( 1 + 2.82e4T + 7.14e8T^{2} \) |
| 61 | \( 1 + 1.92e4T + 8.44e8T^{2} \) |
| 67 | \( 1 + 6.27e3T + 1.35e9T^{2} \) |
| 71 | \( 1 - 6.93e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 5.93e4T + 2.07e9T^{2} \) |
| 79 | \( 1 + 2.84e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + 6.31e4T + 3.93e9T^{2} \) |
| 89 | \( 1 + 1.21e5T + 5.58e9T^{2} \) |
| 97 | \( 1 - 1.23e5T + 8.58e9T^{2} \) |
| 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
−8.933690929389115320103541078773, −7.85038233234227642080297557749, −7.16282134868448081745849148837, −6.18670901746717937684237972550, −5.71080272860481104078044065046, −4.43801257634856405707431582512, −3.59663204337949920989757429663, −1.72287296250598481510471202861, −0.61438544939865893952199441458, 0,
0.61438544939865893952199441458, 1.72287296250598481510471202861, 3.59663204337949920989757429663, 4.43801257634856405707431582512, 5.71080272860481104078044065046, 6.18670901746717937684237972550, 7.16282134868448081745849148837, 7.85038233234227642080297557749, 8.933690929389115320103541078773
