| L(s) = 1 | + 3-s − 5-s + 3·7-s + 9-s + 13-s − 15-s + 8·17-s + 5·19-s + 3·21-s − 6·23-s + 25-s + 27-s − 2·29-s + 7·31-s − 3·35-s + 7·37-s + 39-s + 8·43-s − 45-s + 4·47-s + 2·49-s + 8·51-s − 8·53-s + 5·57-s + 8·59-s + 3·61-s + 3·63-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.447·5-s + 1.13·7-s + 1/3·9-s + 0.277·13-s − 0.258·15-s + 1.94·17-s + 1.14·19-s + 0.654·21-s − 1.25·23-s + 1/5·25-s + 0.192·27-s − 0.371·29-s + 1.25·31-s − 0.507·35-s + 1.15·37-s + 0.160·39-s + 1.21·43-s − 0.149·45-s + 0.583·47-s + 2/7·49-s + 1.12·51-s − 1.09·53-s + 0.662·57-s + 1.04·59-s + 0.384·61-s + 0.377·63-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 188760 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 188760 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(5.530810399\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.530810399\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 5 | \( 1 + T \) |
| 11 | \( 1 \) |
| 13 | \( 1 - T \) |
| good | 7 | \( 1 - 3 T + p T^{2} \) |
| 17 | \( 1 - 8 T + p T^{2} \) |
| 19 | \( 1 - 5 T + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 - 7 T + p T^{2} \) |
| 37 | \( 1 - 7 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 - 4 T + p T^{2} \) |
| 53 | \( 1 + 8 T + p T^{2} \) |
| 59 | \( 1 - 8 T + p T^{2} \) |
| 61 | \( 1 - 3 T + p T^{2} \) |
| 67 | \( 1 - T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 7 T + p T^{2} \) |
| 79 | \( 1 - 17 T + p T^{2} \) |
| 83 | \( 1 + 6 T + p T^{2} \) |
| 89 | \( 1 + p T^{2} \) |
| 97 | \( 1 + 3 T + p T^{2} \) |
| 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
−13.16891994996671, −12.51962726460017, −12.18168362495744, −11.64072092479507, −11.42808226635796, −10.77809247303428, −10.19621429164102, −9.784185494977608, −9.428903296443191, −8.686064253144054, −8.151918885468346, −7.968672957073111, −7.458266957191256, −7.196805477578761, −6.074067355043459, −5.949115256244404, −5.179360601295108, −4.729712207920763, −4.159255846091429, −3.595723211370397, −3.161415527534224, −2.468694694463451, −1.843079085703278, −1.085196014741077, −0.7661922010119318,
0.7661922010119318, 1.085196014741077, 1.843079085703278, 2.468694694463451, 3.161415527534224, 3.595723211370397, 4.159255846091429, 4.729712207920763, 5.179360601295108, 5.949115256244404, 6.074067355043459, 7.196805477578761, 7.458266957191256, 7.968672957073111, 8.151918885468346, 8.686064253144054, 9.428903296443191, 9.784185494977608, 10.19621429164102, 10.77809247303428, 11.42808226635796, 11.64072092479507, 12.18168362495744, 12.51962726460017, 13.16891994996671
