| L(s) = 1 | − 30.6·3-s + 42.6·5-s + 15.7·7-s + 697.·9-s + 262.·11-s + 483.·13-s − 1.30e3·15-s − 1.47e3·17-s + 2.27e3·19-s − 484.·21-s + 259.·23-s − 1.30e3·25-s − 1.39e4·27-s − 6.86e3·29-s − 1.61e3·31-s − 8.04e3·33-s + 672.·35-s − 5.14e3·37-s − 1.48e4·39-s + 7.06e3·41-s + 2.12e4·43-s + 2.97e4·45-s − 1.37e4·47-s − 1.65e4·49-s + 4.53e4·51-s + 6.09e3·53-s + 1.11e4·55-s + ⋯ |
| L(s) = 1 | − 1.96·3-s + 0.762·5-s + 0.121·7-s + 2.87·9-s + 0.653·11-s + 0.792·13-s − 1.49·15-s − 1.24·17-s + 1.44·19-s − 0.239·21-s + 0.102·23-s − 0.418·25-s − 3.68·27-s − 1.51·29-s − 0.300·31-s − 1.28·33-s + 0.0928·35-s − 0.617·37-s − 1.55·39-s + 0.656·41-s + 1.75·43-s + 2.18·45-s − 0.908·47-s − 0.985·49-s + 2.44·51-s + 0.298·53-s + 0.497·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1028 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1028 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 \) |
| 257 | \( 1 - 6.60e4T \) |
| good | 3 | \( 1 + 30.6T + 243T^{2} \) |
| 5 | \( 1 - 42.6T + 3.12e3T^{2} \) |
| 7 | \( 1 - 15.7T + 1.68e4T^{2} \) |
| 11 | \( 1 - 262.T + 1.61e5T^{2} \) |
| 13 | \( 1 - 483.T + 3.71e5T^{2} \) |
| 17 | \( 1 + 1.47e3T + 1.41e6T^{2} \) |
| 19 | \( 1 - 2.27e3T + 2.47e6T^{2} \) |
| 23 | \( 1 - 259.T + 6.43e6T^{2} \) |
| 29 | \( 1 + 6.86e3T + 2.05e7T^{2} \) |
| 31 | \( 1 + 1.61e3T + 2.86e7T^{2} \) |
| 37 | \( 1 + 5.14e3T + 6.93e7T^{2} \) |
| 41 | \( 1 - 7.06e3T + 1.15e8T^{2} \) |
| 43 | \( 1 - 2.12e4T + 1.47e8T^{2} \) |
| 47 | \( 1 + 1.37e4T + 2.29e8T^{2} \) |
| 53 | \( 1 - 6.09e3T + 4.18e8T^{2} \) |
| 59 | \( 1 + 2.43e4T + 7.14e8T^{2} \) |
| 61 | \( 1 + 1.50e4T + 8.44e8T^{2} \) |
| 67 | \( 1 + 3.90e4T + 1.35e9T^{2} \) |
| 71 | \( 1 - 6.04e4T + 1.80e9T^{2} \) |
| 73 | \( 1 - 9.21e3T + 2.07e9T^{2} \) |
| 79 | \( 1 + 3.33e4T + 3.07e9T^{2} \) |
| 83 | \( 1 - 1.88e4T + 3.93e9T^{2} \) |
| 89 | \( 1 - 3.05e4T + 5.58e9T^{2} \) |
| 97 | \( 1 + 2.20e4T + 8.58e9T^{2} \) |
| 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
−9.220560700391687762990552440800, −7.62305944550221169604476637153, −6.82099420879896016447421730581, −6.05139054744489404465843720809, −5.58135519434477608128928355421, −4.66564517265823849598482842293, −3.74334407008027297241115257818, −1.85601163205971324695710725611, −1.12313309138799541757669326525, 0,
1.12313309138799541757669326525, 1.85601163205971324695710725611, 3.74334407008027297241115257818, 4.66564517265823849598482842293, 5.58135519434477608128928355421, 6.05139054744489404465843720809, 6.82099420879896016447421730581, 7.62305944550221169604476637153, 9.220560700391687762990552440800
