| L(s) = 1 | − 2·2-s − 3-s + 2·4-s − 5-s + 2·6-s + 2·7-s − 2·9-s + 2·10-s − 11-s − 2·12-s + 4·13-s − 4·14-s + 15-s − 4·16-s + 2·17-s + 4·18-s − 2·20-s − 2·21-s + 2·22-s − 4·25-s − 8·26-s + 5·27-s + 4·28-s − 2·30-s + 7·31-s + 8·32-s + 33-s + ⋯ |
| L(s) = 1 | − 1.41·2-s − 0.577·3-s + 4-s − 0.447·5-s + 0.816·6-s + 0.755·7-s − 2/3·9-s + 0.632·10-s − 0.301·11-s − 0.577·12-s + 1.10·13-s − 1.06·14-s + 0.258·15-s − 16-s + 0.485·17-s + 0.942·18-s − 0.447·20-s − 0.436·21-s + 0.426·22-s − 4/5·25-s − 1.56·26-s + 0.962·27-s + 0.755·28-s − 0.365·30-s + 1.25·31-s + 1.41·32-s + 0.174·33-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5819 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5819 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.6083685841\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6083685841\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 11 | \( 1 + T \) |
| 23 | \( 1 \) |
| good | 2 | \( 1 + p T + p T^{2} \) |
| 3 | \( 1 + T + p T^{2} \) |
| 5 | \( 1 + T + p T^{2} \) |
| 7 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 - 7 T + p T^{2} \) |
| 37 | \( 1 + 3 T + p T^{2} \) |
| 41 | \( 1 + 8 T + p T^{2} \) |
| 43 | \( 1 - 6 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 - 5 T + p T^{2} \) |
| 61 | \( 1 + 12 T + p T^{2} \) |
| 67 | \( 1 - 7 T + p T^{2} \) |
| 71 | \( 1 + 3 T + p T^{2} \) |
| 73 | \( 1 - 4 T + p T^{2} \) |
| 79 | \( 1 - 10 T + p T^{2} \) |
| 83 | \( 1 - 6 T + p T^{2} \) |
| 89 | \( 1 + 15 T + p T^{2} \) |
| 97 | \( 1 - 7 T + p T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.212973485084645328991374926218, −7.72626049705676692250966013712, −6.87252557874530003114725570070, −6.10199205171023259542319673445, −5.35806526558242506412465562277, −4.54560153511675547350568265865, −3.61425154305823041342611871942, −2.48240948578189648627819728306, −1.42318267584838641544176432302, −0.57357212901825478877549470562,
0.57357212901825478877549470562, 1.42318267584838641544176432302, 2.48240948578189648627819728306, 3.61425154305823041342611871942, 4.54560153511675547350568265865, 5.35806526558242506412465562277, 6.10199205171023259542319673445, 6.87252557874530003114725570070, 7.72626049705676692250966013712, 8.212973485084645328991374926218
