| L(s) = 1 | + 2·5-s − 2·7-s − 4·11-s + 13-s − 2·19-s + 4·23-s − 25-s − 2·31-s − 4·35-s + 10·37-s − 2·41-s + 8·43-s − 3·49-s + 12·53-s − 8·55-s − 12·59-s + 6·61-s + 2·65-s − 6·67-s − 8·71-s − 2·73-s + 8·77-s − 12·79-s + 4·83-s − 14·89-s − 2·91-s − 4·95-s + ⋯ |
| L(s) = 1 | + 0.894·5-s − 0.755·7-s − 1.20·11-s + 0.277·13-s − 0.458·19-s + 0.834·23-s − 1/5·25-s − 0.359·31-s − 0.676·35-s + 1.64·37-s − 0.312·41-s + 1.21·43-s − 3/7·49-s + 1.64·53-s − 1.07·55-s − 1.56·59-s + 0.768·61-s + 0.248·65-s − 0.733·67-s − 0.949·71-s − 0.234·73-s + 0.911·77-s − 1.35·79-s + 0.439·83-s − 1.48·89-s − 0.209·91-s − 0.410·95-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7488 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7488 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 13 | \( 1 - T \) |
| good | 5 | \( 1 - 2 T + p T^{2} \) |
| 7 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 + 2 T + p T^{2} \) |
| 37 | \( 1 - 10 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 12 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 - 6 T + p T^{2} \) |
| 67 | \( 1 + 6 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 + 12 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 + 14 T + p T^{2} \) |
| 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
−7.49026915499148598420863229304, −6.82611641620904034974269009642, −5.94660273142565104367971439251, −5.69655062923021187304922056070, −4.77219915335120526458588257092, −3.93999038741216645545048796411, −2.87117150112182190841690637877, −2.45734264446474454889575501395, −1.30745574631339359324468269776, 0,
1.30745574631339359324468269776, 2.45734264446474454889575501395, 2.87117150112182190841690637877, 3.93999038741216645545048796411, 4.77219915335120526458588257092, 5.69655062923021187304922056070, 5.94660273142565104367971439251, 6.82611641620904034974269009642, 7.49026915499148598420863229304
