| L(s) = 1 | − 5-s + 4·11-s − 13-s + 2·17-s + 4·19-s + 8·23-s + 25-s − 6·29-s + 10·37-s − 6·41-s − 4·43-s − 7·49-s + 6·53-s − 4·55-s − 12·59-s + 10·61-s + 65-s − 12·67-s − 12·71-s − 10·73-s − 8·79-s − 16·83-s − 2·85-s − 14·89-s − 4·95-s − 2·97-s + 101-s + ⋯ |
| L(s) = 1 | − 0.447·5-s + 1.20·11-s − 0.277·13-s + 0.485·17-s + 0.917·19-s + 1.66·23-s + 1/5·25-s − 1.11·29-s + 1.64·37-s − 0.937·41-s − 0.609·43-s − 49-s + 0.824·53-s − 0.539·55-s − 1.56·59-s + 1.28·61-s + 0.124·65-s − 1.46·67-s − 1.42·71-s − 1.17·73-s − 0.900·79-s − 1.75·83-s − 0.216·85-s − 1.48·89-s − 0.410·95-s − 0.203·97-s + 0.0995·101-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 37440 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 37440 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 13 | \( 1 + T \) |
| good | 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 - 8 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 - 10 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 + 16 T + p T^{2} \) |
| 89 | \( 1 + 14 T + p T^{2} \) |
| 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
−15.02490435996190, −14.69608149242762, −14.20516264221472, −13.53889754375319, −12.96457899760908, −12.60513250850308, −11.70232877178177, −11.59472298209372, −11.15598299345303, −10.27169780369708, −9.824831610022136, −9.183324135571067, −8.854990721462891, −8.148077845076850, −7.341025980473864, −7.232284864484462, −6.451008120546908, −5.798814436007241, −5.191816946585601, −4.507672216470851, −3.968008937137470, −3.177064970195747, −2.828290126892985, −1.548443905308889, −1.151337481853590, 0,
1.151337481853590, 1.548443905308889, 2.828290126892985, 3.177064970195747, 3.968008937137470, 4.507672216470851, 5.191816946585601, 5.798814436007241, 6.451008120546908, 7.232284864484462, 7.341025980473864, 8.148077845076850, 8.854990721462891, 9.183324135571067, 9.824831610022136, 10.27169780369708, 11.15598299345303, 11.59472298209372, 11.70232877178177, 12.60513250850308, 12.96457899760908, 13.53889754375319, 14.20516264221472, 14.69608149242762, 15.02490435996190
