L(s) = 1 | + 2·5-s − 4·7-s + 11-s − 2·13-s + 2·17-s + 8·23-s − 25-s + 6·29-s + 8·31-s − 8·35-s + 6·37-s + 2·41-s + 8·47-s + 9·49-s − 6·53-s + 2·55-s − 4·59-s + 6·61-s − 4·65-s + 4·67-s − 14·73-s − 4·77-s + 4·79-s + 12·83-s + 4·85-s + 6·89-s + 8·91-s + ⋯ |
L(s) = 1 | + 0.894·5-s − 1.51·7-s + 0.301·11-s − 0.554·13-s + 0.485·17-s + 1.66·23-s − 1/5·25-s + 1.11·29-s + 1.43·31-s − 1.35·35-s + 0.986·37-s + 0.312·41-s + 1.16·47-s + 9/7·49-s − 0.824·53-s + 0.269·55-s − 0.520·59-s + 0.768·61-s − 0.496·65-s + 0.488·67-s − 1.63·73-s − 0.455·77-s + 0.450·79-s + 1.31·83-s + 0.433·85-s + 0.635·89-s + 0.838·91-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1584 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1584 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.725905832\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.725905832\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 11 | \( 1 - T \) |
good | 5 | \( 1 - 2 T + p T^{2} \) |
| 7 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 - 8 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 - 6 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 14 T + p T^{2} \) |
| 79 | \( 1 - 4 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 - 2 T + p T^{2} \) |
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
−9.522485991677101039441768614921, −8.904851262578921543189533962233, −7.75845614267043749557874942232, −6.75687248691737052765776352788, −6.30502880333235153369727352923, −5.44733142268678909820409381105, −4.42115778606690496889098125190, −3.17498211380366678113802704664, −2.53280394883161174744241592337, −0.938050277873206229946250423607,
0.938050277873206229946250423607, 2.53280394883161174744241592337, 3.17498211380366678113802704664, 4.42115778606690496889098125190, 5.44733142268678909820409381105, 6.30502880333235153369727352923, 6.75687248691737052765776352788, 7.75845614267043749557874942232, 8.904851262578921543189533962233, 9.522485991677101039441768614921