| L(s) = 1 | + 5-s + 7-s + 4·11-s + 2·13-s − 2·17-s + 4·19-s + 8·23-s + 25-s + 6·29-s − 8·31-s + 35-s + 2·37-s − 2·41-s + 12·43-s + 8·47-s + 49-s + 6·53-s + 4·55-s + 4·59-s + 2·61-s + 2·65-s − 12·67-s − 8·71-s − 14·73-s + 4·77-s + 12·83-s − 2·85-s + ⋯ |
| L(s) = 1 | + 0.447·5-s + 0.377·7-s + 1.20·11-s + 0.554·13-s − 0.485·17-s + 0.917·19-s + 1.66·23-s + 1/5·25-s + 1.11·29-s − 1.43·31-s + 0.169·35-s + 0.328·37-s − 0.312·41-s + 1.82·43-s + 1.16·47-s + 1/7·49-s + 0.824·53-s + 0.539·55-s + 0.520·59-s + 0.256·61-s + 0.248·65-s − 1.46·67-s − 0.949·71-s − 1.63·73-s + 0.455·77-s + 1.31·83-s − 0.216·85-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 20160 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 20160 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.568887725\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.568887725\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 7 | \( 1 - T \) |
| good | 11 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 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 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 - 12 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 + 14 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 + 2 T + p T^{2} \) |
| 97 | \( 1 - 10 T + p T^{2} \) |
| 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
−15.66801461668974, −14.96772533140457, −14.57230995141674, −14.04335366487783, −13.50501445515472, −13.02262056972428, −12.32876373094897, −11.76038455534062, −11.25309600040401, −10.71903309617147, −10.17109091807806, −9.283643285582804, −8.952868214964111, −8.649298499694827, −7.446664462844097, −7.267880856873553, −6.435397181953344, −5.892086777844393, −5.246465035980816, −4.523209275913253, −3.896850243745391, −3.123152696208963, −2.366004526010983, −1.381442210075654, −0.8917103692719907,
0.8917103692719907, 1.381442210075654, 2.366004526010983, 3.123152696208963, 3.896850243745391, 4.523209275913253, 5.246465035980816, 5.892086777844393, 6.435397181953344, 7.267880856873553, 7.446664462844097, 8.649298499694827, 8.952868214964111, 9.283643285582804, 10.17109091807806, 10.71903309617147, 11.25309600040401, 11.76038455534062, 12.32876373094897, 13.02262056972428, 13.50501445515472, 14.04335366487783, 14.57230995141674, 14.96772533140457, 15.66801461668974
