L(s) = 1 | + 2·5-s + 11-s + 2·13-s + 6·17-s + 4·19-s − 23-s − 25-s − 6·29-s + 2·37-s − 6·41-s + 4·43-s − 7·49-s + 2·53-s + 2·55-s + 4·59-s + 2·61-s + 4·65-s − 8·67-s − 14·73-s − 4·83-s + 12·85-s − 6·89-s + 8·95-s − 6·97-s + 101-s + 103-s + 107-s + ⋯ |
L(s) = 1 | + 0.894·5-s + 0.301·11-s + 0.554·13-s + 1.45·17-s + 0.917·19-s − 0.208·23-s − 1/5·25-s − 1.11·29-s + 0.328·37-s − 0.937·41-s + 0.609·43-s − 49-s + 0.274·53-s + 0.269·55-s + 0.520·59-s + 0.256·61-s + 0.496·65-s − 0.977·67-s − 1.63·73-s − 0.439·83-s + 1.30·85-s − 0.635·89-s + 0.820·95-s − 0.609·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 145728 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 145728 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 11 | \( 1 - T \) |
| 23 | \( 1 + T \) |
good | 5 | \( 1 - 2 T + p T^{2} \) |
| 7 | \( 1 + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 2 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 + 8 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 14 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 + 6 T + p T^{2} \) |
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
−13.58609061767861, −13.19908907235346, −12.75388614447355, −12.13987066481856, −11.65870783876124, −11.35551114285997, −10.64035120812793, −10.12795022847955, −9.787340741763902, −9.390041877624154, −8.845332351615535, −8.290269090635936, −7.686682711703275, −7.328364298236808, −6.690972089919762, −6.011643534617583, −5.726935833984813, −5.329784287508257, −4.640012976110543, −3.902918102834675, −3.424537848015804, −2.899691170837003, −2.125615865998712, −1.458231200325434, −1.100988085911184, 0,
1.100988085911184, 1.458231200325434, 2.125615865998712, 2.899691170837003, 3.424537848015804, 3.902918102834675, 4.640012976110543, 5.329784287508257, 5.726935833984813, 6.011643534617583, 6.690972089919762, 7.328364298236808, 7.686682711703275, 8.290269090635936, 8.845332351615535, 9.390041877624154, 9.787340741763902, 10.12795022847955, 10.64035120812793, 11.35551114285997, 11.65870783876124, 12.13987066481856, 12.75388614447355, 13.19908907235346, 13.58609061767861