| L(s) = 1 | + 5-s + 7-s + 4·11-s − 6·13-s − 2·17-s + 25-s − 6·29-s − 8·31-s + 35-s − 10·37-s − 2·41-s − 4·43-s + 8·47-s + 49-s + 2·53-s + 4·55-s − 8·59-s − 14·61-s − 6·65-s + 12·67-s − 16·71-s + 2·73-s + 4·77-s + 8·79-s + 8·83-s − 2·85-s − 10·89-s + ⋯ |
| L(s) = 1 | + 0.447·5-s + 0.377·7-s + 1.20·11-s − 1.66·13-s − 0.485·17-s + 1/5·25-s − 1.11·29-s − 1.43·31-s + 0.169·35-s − 1.64·37-s − 0.312·41-s − 0.609·43-s + 1.16·47-s + 1/7·49-s + 0.274·53-s + 0.539·55-s − 1.04·59-s − 1.79·61-s − 0.744·65-s + 1.46·67-s − 1.89·71-s + 0.234·73-s + 0.455·77-s + 0.900·79-s + 0.878·83-s − 0.216·85-s − 1.05·89-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5040 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5040 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 7 | \( 1 - T \) |
| good | 11 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 + 6 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 + 10 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 - 2 T + p T^{2} \) |
| 59 | \( 1 + 8 T + p T^{2} \) |
| 61 | \( 1 + 14 T + p T^{2} \) |
| 67 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 + 16 T + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 - 8 T + p T^{2} \) |
| 89 | \( 1 + 10 T + p T^{2} \) |
| 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
−7.69537665846501814851151036032, −7.18442932220920716960195375619, −6.53503531475229539424605190008, −5.60334584590381835154023547959, −4.99605143965867724115284774422, −4.18930851846167852256621728779, −3.33062511491831582375696335365, −2.19590154838649435040150777594, −1.57587941250931515532871434236, 0,
1.57587941250931515532871434236, 2.19590154838649435040150777594, 3.33062511491831582375696335365, 4.18930851846167852256621728779, 4.99605143965867724115284774422, 5.60334584590381835154023547959, 6.53503531475229539424605190008, 7.18442932220920716960195375619, 7.69537665846501814851151036032
