L(s) = 1 | + 2-s + 4-s + 2·5-s − 7-s + 8-s + 2·10-s + 2·11-s + 2·13-s − 14-s + 16-s + 4·17-s + 19-s + 2·20-s + 2·22-s − 25-s + 2·26-s − 28-s + 6·29-s − 10·31-s + 32-s + 4·34-s − 2·35-s + 38-s + 2·40-s + 6·41-s − 4·43-s + 2·44-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 1/2·4-s + 0.894·5-s − 0.377·7-s + 0.353·8-s + 0.632·10-s + 0.603·11-s + 0.554·13-s − 0.267·14-s + 1/4·16-s + 0.970·17-s + 0.229·19-s + 0.447·20-s + 0.426·22-s − 1/5·25-s + 0.392·26-s − 0.188·28-s + 1.11·29-s − 1.79·31-s + 0.176·32-s + 0.685·34-s − 0.338·35-s + 0.162·38-s + 0.316·40-s + 0.937·41-s − 0.609·43-s + 0.301·44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2394 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2394 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(3.566965991\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.566965991\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - T \) |
| 3 | \( 1 \) |
| 7 | \( 1 + T \) |
| 19 | \( 1 - T \) |
good | 5 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + 10 T + p T^{2} \) |
| 37 | \( 1 + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + 6 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 + 2 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 + 6 T + p T^{2} \) |
| 79 | \( 1 - 16 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 + 10 T + p T^{2} \) |
| 97 | \( 1 + 12 T + p T^{2} \) |
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
−9.078168737525253415228268729002, −8.170795419592986651635190702980, −7.18996824626718531971644901105, −6.47826890764947884762493111554, −5.76776105743104111407787752275, −5.21253501588227766714841283002, −4.01592661411596137857298468735, −3.33132084738412322385973884213, −2.24695454904725672678180055001, −1.20242098191103416224247454645,
1.20242098191103416224247454645, 2.24695454904725672678180055001, 3.33132084738412322385973884213, 4.01592661411596137857298468735, 5.21253501588227766714841283002, 5.76776105743104111407787752275, 6.47826890764947884762493111554, 7.18996824626718531971644901105, 8.170795419592986651635190702980, 9.078168737525253415228268729002