| L(s) = 1 | + 5-s − 4·7-s + 4·11-s + 2·13-s + 2·17-s + 2·19-s + 2·23-s + 25-s + 6·29-s − 4·31-s − 4·35-s + 8·37-s + 8·41-s − 43-s − 6·47-s + 9·49-s + 6·53-s + 4·55-s − 6·61-s + 2·65-s − 12·67-s − 8·71-s − 4·73-s − 16·77-s + 4·79-s + 6·83-s + 2·85-s + ⋯ |
| L(s) = 1 | + 0.447·5-s − 1.51·7-s + 1.20·11-s + 0.554·13-s + 0.485·17-s + 0.458·19-s + 0.417·23-s + 1/5·25-s + 1.11·29-s − 0.718·31-s − 0.676·35-s + 1.31·37-s + 1.24·41-s − 0.152·43-s − 0.875·47-s + 9/7·49-s + 0.824·53-s + 0.539·55-s − 0.768·61-s + 0.248·65-s − 1.46·67-s − 0.949·71-s − 0.468·73-s − 1.82·77-s + 0.450·79-s + 0.658·83-s + 0.216·85-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 123840 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 123840 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 43 | \( 1 + T \) |
| good | 7 | \( 1 + 4 T + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 - 2 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 - 8 T + p T^{2} \) |
| 41 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 + 6 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 + 6 T + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 + 4 T + p T^{2} \) |
| 79 | \( 1 - 4 T + p T^{2} \) |
| 83 | \( 1 - 6 T + p T^{2} \) |
| 89 | \( 1 - 2 T + p T^{2} \) |
| 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
−13.65827096396393, −13.32682017532098, −12.83668096260754, −12.40215378388367, −11.87156172326038, −11.47242623470183, −10.77174045396678, −10.32805991315690, −9.808673538848649, −9.338409736005074, −9.092956705441770, −8.568989490443814, −7.745777668756208, −7.308228007128688, −6.641256903013109, −6.262033831545413, −5.993857374580455, −5.321705061041998, −4.566931657106872, −3.972659271943055, −3.460448158089166, −2.934548573000831, −2.415826580426133, −1.322809902524805, −1.038675432961544, 0,
1.038675432961544, 1.322809902524805, 2.415826580426133, 2.934548573000831, 3.460448158089166, 3.972659271943055, 4.566931657106872, 5.321705061041998, 5.993857374580455, 6.262033831545413, 6.641256903013109, 7.308228007128688, 7.745777668756208, 8.568989490443814, 9.092956705441770, 9.338409736005074, 9.808673538848649, 10.32805991315690, 10.77174045396678, 11.47242623470183, 11.87156172326038, 12.40215378388367, 12.83668096260754, 13.32682017532098, 13.65827096396393
