| L(s) = 1 | − 4·7-s + 13-s − 3·19-s − 4·23-s + 29-s − 8·31-s + 37-s − 41-s − 6·43-s − 11·47-s + 9·49-s − 3·53-s − 10·59-s + 4·61-s − 13·67-s − 9·71-s − 3·79-s + 2·83-s − 10·89-s − 4·91-s + 8·97-s + 101-s + 103-s + 107-s + 109-s + 113-s + ⋯ |
| L(s) = 1 | − 1.51·7-s + 0.277·13-s − 0.688·19-s − 0.834·23-s + 0.185·29-s − 1.43·31-s + 0.164·37-s − 0.156·41-s − 0.914·43-s − 1.60·47-s + 9/7·49-s − 0.412·53-s − 1.30·59-s + 0.512·61-s − 1.58·67-s − 1.06·71-s − 0.337·79-s + 0.219·83-s − 1.05·89-s − 0.419·91-s + 0.812·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + 0.0940·113-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 93600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 93600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| 13 | \( 1 - T \) |
| good | 7 | \( 1 + 4 T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 + 3 T + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 - T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 - T + p T^{2} \) |
| 41 | \( 1 + T + p T^{2} \) |
| 43 | \( 1 + 6 T + p T^{2} \) |
| 47 | \( 1 + 11 T + p T^{2} \) |
| 53 | \( 1 + 3 T + p T^{2} \) |
| 59 | \( 1 + 10 T + p T^{2} \) |
| 61 | \( 1 - 4 T + p T^{2} \) |
| 67 | \( 1 + 13 T + p T^{2} \) |
| 71 | \( 1 + 9 T + p T^{2} \) |
| 73 | \( 1 + p T^{2} \) |
| 79 | \( 1 + 3 T + p T^{2} \) |
| 83 | \( 1 - 2 T + p T^{2} \) |
| 89 | \( 1 + 10 T + p T^{2} \) |
| 97 | \( 1 - 8 T + p T^{2} \) |
| 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
−14.34056367099119, −13.71028972354734, −13.26779971793037, −12.73824488871514, −12.64100421144829, −11.83213699568112, −11.45125385398723, −10.78857717814954, −10.24915625398529, −9.919107950376543, −9.400084272673895, −8.821645784659519, −8.484075068014276, −7.647619916772879, −7.299598287305903, −6.534955788334269, −6.231188450507897, −5.840566767707216, −5.021912471785629, −4.457042136773309, −3.688228925831710, −3.387290685891195, −2.747228652926779, −1.981847335162921, −1.324436251757097, 0, 0,
1.324436251757097, 1.981847335162921, 2.747228652926779, 3.387290685891195, 3.688228925831710, 4.457042136773309, 5.021912471785629, 5.840566767707216, 6.231188450507897, 6.534955788334269, 7.299598287305903, 7.647619916772879, 8.484075068014276, 8.821645784659519, 9.400084272673895, 9.919107950376543, 10.24915625398529, 10.78857717814954, 11.45125385398723, 11.83213699568112, 12.64100421144829, 12.73824488871514, 13.26779971793037, 13.71028972354734, 14.34056367099119
