| L(s) = 1 | + 2·2-s + 2·4-s + 5-s + 2·10-s − 11-s + 3·13-s − 4·16-s + 3·17-s + 6·19-s + 2·20-s − 2·22-s + 4·23-s + 25-s + 6·26-s + 29-s + 6·31-s − 8·32-s + 6·34-s + 12·38-s − 6·41-s − 6·43-s − 2·44-s + 8·46-s + 9·47-s + 2·50-s + 6·52-s + 10·53-s + ⋯ |
| L(s) = 1 | + 1.41·2-s + 4-s + 0.447·5-s + 0.632·10-s − 0.301·11-s + 0.832·13-s − 16-s + 0.727·17-s + 1.37·19-s + 0.447·20-s − 0.426·22-s + 0.834·23-s + 1/5·25-s + 1.17·26-s + 0.185·29-s + 1.07·31-s − 1.41·32-s + 1.02·34-s + 1.94·38-s − 0.937·41-s − 0.914·43-s − 0.301·44-s + 1.17·46-s + 1.31·47-s + 0.282·50-s + 0.832·52-s + 1.37·53-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2205 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2205 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.287238192\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.287238192\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 - T \) |
| 7 | \( 1 \) |
| good | 2 | \( 1 - p T + p T^{2} \) |
| 11 | \( 1 + T + p T^{2} \) |
| 13 | \( 1 - 3 T + p T^{2} \) |
| 17 | \( 1 - 3 T + p T^{2} \) |
| 19 | \( 1 - 6 T + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 - T + p T^{2} \) |
| 31 | \( 1 - 6 T + p T^{2} \) |
| 37 | \( 1 + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 + 6 T + p T^{2} \) |
| 47 | \( 1 - 9 T + p T^{2} \) |
| 53 | \( 1 - 10 T + p T^{2} \) |
| 59 | \( 1 - 6 T + p T^{2} \) |
| 61 | \( 1 + p T^{2} \) |
| 67 | \( 1 + 14 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 + T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 + 12 T + p T^{2} \) |
| 97 | \( 1 + 15 T + p T^{2} \) |
| 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
−9.050119098227053621632888047745, −8.252623699886366464992405726957, −7.18674908490146366638229016691, −6.50048591700478498626320913392, −5.54795317274704426653960185894, −5.25756591986209044151466901261, −4.20509313138730818892241036590, −3.31721807269730857488302455154, −2.65178092612008457940193320214, −1.20406663212713068028112418239,
1.20406663212713068028112418239, 2.65178092612008457940193320214, 3.31721807269730857488302455154, 4.20509313138730818892241036590, 5.25756591986209044151466901261, 5.54795317274704426653960185894, 6.50048591700478498626320913392, 7.18674908490146366638229016691, 8.252623699886366464992405726957, 9.050119098227053621632888047745
