| L(s) = 1 | + 2-s + 4-s − 7-s + 8-s − 2·11-s + 6·13-s − 14-s + 16-s + 5·17-s + 6·19-s − 2·22-s − 3·23-s + 6·26-s − 28-s − 6·29-s − 4·31-s + 32-s + 5·34-s − 4·37-s + 6·38-s − 41-s + 7·43-s − 2·44-s − 3·46-s + 2·47-s + 49-s + 6·52-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s − 0.377·7-s + 0.353·8-s − 0.603·11-s + 1.66·13-s − 0.267·14-s + 1/4·16-s + 1.21·17-s + 1.37·19-s − 0.426·22-s − 0.625·23-s + 1.17·26-s − 0.188·28-s − 1.11·29-s − 0.718·31-s + 0.176·32-s + 0.857·34-s − 0.657·37-s + 0.973·38-s − 0.156·41-s + 1.06·43-s − 0.301·44-s − 0.442·46-s + 0.291·47-s + 1/7·49-s + 0.832·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9450 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.641789362\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.641789362\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 + T \) |
| good | 11 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 - 6 T + p T^{2} \) |
| 17 | \( 1 - 5 T + p T^{2} \) |
| 19 | \( 1 - 6 T + p T^{2} \) |
| 23 | \( 1 + 3 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 + 4 T + p T^{2} \) |
| 41 | \( 1 + T + p T^{2} \) |
| 43 | \( 1 - 7 T + p T^{2} \) |
| 47 | \( 1 - 2 T + p T^{2} \) |
| 53 | \( 1 + 2 T + p T^{2} \) |
| 59 | \( 1 - 7 T + p T^{2} \) |
| 61 | \( 1 - 13 T + p T^{2} \) |
| 67 | \( 1 - 9 T + p T^{2} \) |
| 71 | \( 1 + 7 T + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 - 10 T + p T^{2} \) |
| 83 | \( 1 - 17 T + p T^{2} \) |
| 89 | \( 1 + 11 T + p T^{2} \) |
| 97 | \( 1 - 8 T + p T^{2} \) |
| 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
−7.63288494786743994150624870740, −6.98826100534599959706852137291, −6.13066242362709650280178170365, −5.55132915752760577496201551775, −5.21234545806482512783980419049, −3.87343688582806334478284091069, −3.64832171229595958134316218279, −2.84619944967830233427995831705, −1.79169800973114878280382843910, −0.848027889714959850393511257301,
0.848027889714959850393511257301, 1.79169800973114878280382843910, 2.84619944967830233427995831705, 3.64832171229595958134316218279, 3.87343688582806334478284091069, 5.21234545806482512783980419049, 5.55132915752760577496201551775, 6.13066242362709650280178170365, 6.98826100534599959706852137291, 7.63288494786743994150624870740
