| L(s) = 1 | + 5-s − 4·7-s − 2·13-s + 17-s + 23-s + 25-s − 6·29-s + 31-s − 4·35-s + 4·37-s + 2·41-s + 2·43-s − 9·47-s + 9·49-s − 9·53-s + 6·59-s − 61-s − 2·65-s + 4·67-s − 4·73-s + 11·79-s + 8·83-s + 85-s + 16·89-s + 8·91-s + 16·97-s + 101-s + ⋯ |
| L(s) = 1 | + 0.447·5-s − 1.51·7-s − 0.554·13-s + 0.242·17-s + 0.208·23-s + 1/5·25-s − 1.11·29-s + 0.179·31-s − 0.676·35-s + 0.657·37-s + 0.312·41-s + 0.304·43-s − 1.31·47-s + 9/7·49-s − 1.23·53-s + 0.781·59-s − 0.128·61-s − 0.248·65-s + 0.488·67-s − 0.468·73-s + 1.23·79-s + 0.878·83-s + 0.108·85-s + 1.69·89-s + 0.838·91-s + 1.62·97-s + 0.0995·101-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 43560 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43560 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 11 | \( 1 \) |
| good | 7 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 - T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 - T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 - T + p T^{2} \) |
| 37 | \( 1 - 4 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 - 2 T + p T^{2} \) |
| 47 | \( 1 + 9 T + p T^{2} \) |
| 53 | \( 1 + 9 T + p T^{2} \) |
| 59 | \( 1 - 6 T + p T^{2} \) |
| 61 | \( 1 + T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 4 T + p T^{2} \) |
| 79 | \( 1 - 11 T + p T^{2} \) |
| 83 | \( 1 - 8 T + p T^{2} \) |
| 89 | \( 1 - 16 T + p T^{2} \) |
| 97 | \( 1 - 16 T + p T^{2} \) |
| 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
−14.81266416451757, −14.55141583934004, −13.80501084095522, −13.29300790834470, −12.88599065973254, −12.55164790703819, −11.89996088883411, −11.30516825582838, −10.69338838279958, −10.09902480107810, −9.655738776272543, −9.321485284437017, −8.800273575263223, −7.879895148411966, −7.525654745952626, −6.689580969020134, −6.421615847545570, −5.808100691153549, −5.191093955632294, −4.549333272111602, −3.675009537684099, −3.258847503060367, −2.554938316283019, −1.908593643682142, −0.8579152093345204, 0,
0.8579152093345204, 1.908593643682142, 2.554938316283019, 3.258847503060367, 3.675009537684099, 4.549333272111602, 5.191093955632294, 5.808100691153549, 6.421615847545570, 6.689580969020134, 7.525654745952626, 7.879895148411966, 8.800273575263223, 9.321485284437017, 9.655738776272543, 10.09902480107810, 10.69338838279958, 11.30516825582838, 11.89996088883411, 12.55164790703819, 12.88599065973254, 13.29300790834470, 13.80501084095522, 14.55141583934004, 14.81266416451757
