L(s) = 1 | + 1.09·2-s + 1.16·3-s − 0.798·4-s + 1.27·6-s + 4.08·7-s − 3.06·8-s − 1.65·9-s + 3.43·11-s − 0.927·12-s − 3.30·13-s + 4.47·14-s − 1.76·16-s − 3.47·17-s − 1.81·18-s + 0.926·19-s + 4.73·21-s + 3.76·22-s − 4.34·23-s − 3.56·24-s − 3.61·26-s − 5.40·27-s − 3.26·28-s + 4.51·29-s + 5.56·31-s + 4.20·32-s + 3.99·33-s − 3.80·34-s + ⋯ |
L(s) = 1 | + 0.774·2-s + 0.670·3-s − 0.399·4-s + 0.519·6-s + 1.54·7-s − 1.08·8-s − 0.550·9-s + 1.03·11-s − 0.267·12-s − 0.915·13-s + 1.19·14-s − 0.441·16-s − 0.841·17-s − 0.426·18-s + 0.212·19-s + 1.03·21-s + 0.803·22-s − 0.906·23-s − 0.726·24-s − 0.709·26-s − 1.03·27-s − 0.616·28-s + 0.837·29-s + 1.00·31-s + 0.742·32-s + 0.694·33-s − 0.652·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6275 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6275 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(3.556676227\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.556676227\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 \) |
| 251 | \( 1 - T \) |
good | 2 | \( 1 - 1.09T + 2T^{2} \) |
| 3 | \( 1 - 1.16T + 3T^{2} \) |
| 7 | \( 1 - 4.08T + 7T^{2} \) |
| 11 | \( 1 - 3.43T + 11T^{2} \) |
| 13 | \( 1 + 3.30T + 13T^{2} \) |
| 17 | \( 1 + 3.47T + 17T^{2} \) |
| 19 | \( 1 - 0.926T + 19T^{2} \) |
| 23 | \( 1 + 4.34T + 23T^{2} \) |
| 29 | \( 1 - 4.51T + 29T^{2} \) |
| 31 | \( 1 - 5.56T + 31T^{2} \) |
| 37 | \( 1 - 8.98T + 37T^{2} \) |
| 41 | \( 1 - 3.94T + 41T^{2} \) |
| 43 | \( 1 - 7.51T + 43T^{2} \) |
| 47 | \( 1 - 5.92T + 47T^{2} \) |
| 53 | \( 1 - 1.64T + 53T^{2} \) |
| 59 | \( 1 + 10.8T + 59T^{2} \) |
| 61 | \( 1 - 13.3T + 61T^{2} \) |
| 67 | \( 1 - 5.72T + 67T^{2} \) |
| 71 | \( 1 - 0.225T + 71T^{2} \) |
| 73 | \( 1 - 15.9T + 73T^{2} \) |
| 79 | \( 1 - 6.61T + 79T^{2} \) |
| 83 | \( 1 + 13.0T + 83T^{2} \) |
| 89 | \( 1 - 8.08T + 89T^{2} \) |
| 97 | \( 1 + 13.1T + 97T^{2} \) |
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
−8.204790707357835245408933118091, −7.48323450120211131595231526173, −6.46102332306191832632908218568, −5.77431561148031073653388006889, −4.97737753006808188678271134149, −4.34765509845406215831162829455, −3.91842797564082907460135161309, −2.71539963538040020618742951177, −2.20319131203726333995845859876, −0.857152729032132306349271978924,
0.857152729032132306349271978924, 2.20319131203726333995845859876, 2.71539963538040020618742951177, 3.91842797564082907460135161309, 4.34765509845406215831162829455, 4.97737753006808188678271134149, 5.77431561148031073653388006889, 6.46102332306191832632908218568, 7.48323450120211131595231526173, 8.204790707357835245408933118091