| L(s) = 1 | + 3-s − 5-s − 4·7-s + 9-s + 2·11-s − 15-s + 2·17-s − 4·21-s − 23-s + 25-s + 27-s − 4·29-s + 2·33-s + 4·35-s + 10·37-s + 6·41-s − 2·43-s − 45-s − 12·47-s + 9·49-s + 2·51-s + 6·53-s − 2·55-s − 12·59-s − 14·61-s − 4·63-s − 2·67-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.447·5-s − 1.51·7-s + 1/3·9-s + 0.603·11-s − 0.258·15-s + 0.485·17-s − 0.872·21-s − 0.208·23-s + 1/5·25-s + 0.192·27-s − 0.742·29-s + 0.348·33-s + 0.676·35-s + 1.64·37-s + 0.937·41-s − 0.304·43-s − 0.149·45-s − 1.75·47-s + 9/7·49-s + 0.280·51-s + 0.824·53-s − 0.269·55-s − 1.56·59-s − 1.79·61-s − 0.503·63-s − 0.244·67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5520 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 5 | \( 1 + T \) |
| 23 | \( 1 + T \) |
| good | 7 | \( 1 + 4 T + p T^{2} \) |
| 11 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 4 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 - 10 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 + 2 T + p T^{2} \) |
| 47 | \( 1 + 12 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 + 14 T + p T^{2} \) |
| 67 | \( 1 + 2 T + p T^{2} \) |
| 71 | \( 1 - 2 T + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 + 8 T + p T^{2} \) |
| 89 | \( 1 + 8 T + p T^{2} \) |
| 97 | \( 1 + p T^{2} \) |
| 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
−7.73856835628901356556640507177, −7.18305423154109033651224150361, −6.34403425269286096783355345684, −5.90260248226360242464241628952, −4.67628553593983312608557996774, −3.89463425079179803914861429740, −3.28689859692490364257547380800, −2.61049944367168359356905830543, −1.32156945980241101621888183024, 0,
1.32156945980241101621888183024, 2.61049944367168359356905830543, 3.28689859692490364257547380800, 3.89463425079179803914861429740, 4.67628553593983312608557996774, 5.90260248226360242464241628952, 6.34403425269286096783355345684, 7.18305423154109033651224150361, 7.73856835628901356556640507177
