L(s) = 1 | − 3-s − 5-s − 7-s + 9-s − 4·11-s + 2·13-s + 15-s + 2·17-s − 4·19-s + 21-s + 25-s − 27-s + 2·29-s + 8·31-s + 4·33-s + 35-s + 2·37-s − 2·39-s + 2·41-s − 4·43-s − 45-s + 49-s − 2·51-s + 10·53-s + 4·55-s + 4·57-s + 12·59-s + ⋯ |
L(s) = 1 | − 0.577·3-s − 0.447·5-s − 0.377·7-s + 1/3·9-s − 1.20·11-s + 0.554·13-s + 0.258·15-s + 0.485·17-s − 0.917·19-s + 0.218·21-s + 1/5·25-s − 0.192·27-s + 0.371·29-s + 1.43·31-s + 0.696·33-s + 0.169·35-s + 0.328·37-s − 0.320·39-s + 0.312·41-s − 0.609·43-s − 0.149·45-s + 1/7·49-s − 0.280·51-s + 1.37·53-s + 0.539·55-s + 0.529·57-s + 1.56·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6720 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6720 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 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 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 10 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 + 6 T + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 6 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 - 2 T + p T^{2} \) |
| 97 | \( 1 + 14 T + p T^{2} \) |
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
−7.61465113149376131388775328245, −6.90940038220234233328140871933, −6.16458557561649791843039843687, −5.59613722468513539505922609511, −4.74222449423502780904305824189, −4.11664567523245606800028850898, −3.15828342333522043788763466295, −2.38873475787086144556332957685, −1.06885418828923509331016753538, 0,
1.06885418828923509331016753538, 2.38873475787086144556332957685, 3.15828342333522043788763466295, 4.11664567523245606800028850898, 4.74222449423502780904305824189, 5.59613722468513539505922609511, 6.16458557561649791843039843687, 6.90940038220234233328140871933, 7.61465113149376131388775328245