L(s) = 1 | + 2.64·2-s + 5.00·4-s − 5-s + 1.64·7-s + 7.93·8-s − 2.64·10-s − 0.354·11-s − 0.354·13-s + 4.35·14-s + 11.0·16-s + 4·17-s − 19-s − 5.00·20-s − 0.937·22-s − 9.29·23-s + 25-s − 0.937·26-s + 8.22·28-s − 8.93·29-s + 6·31-s + 13.2·32-s + 10.5·34-s − 1.64·35-s + 3.64·37-s − 2.64·38-s − 7.93·40-s + 9.64·41-s + ⋯ |
L(s) = 1 | + 1.87·2-s + 2.50·4-s − 0.447·5-s + 0.622·7-s + 2.80·8-s − 0.836·10-s − 0.106·11-s − 0.0982·13-s + 1.16·14-s + 2.75·16-s + 0.970·17-s − 0.229·19-s − 1.11·20-s − 0.199·22-s − 1.93·23-s + 0.200·25-s − 0.183·26-s + 1.55·28-s − 1.65·29-s + 1.07·31-s + 2.33·32-s + 1.81·34-s − 0.278·35-s + 0.599·37-s − 0.429·38-s − 1.25·40-s + 1.50·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 855 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 855 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(4.719630185\) |
\(L(\frac12)\) |
\(\approx\) |
\(4.719630185\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 5 | \( 1 + T \) |
| 19 | \( 1 + T \) |
good | 2 | \( 1 - 2.64T + 2T^{2} \) |
| 7 | \( 1 - 1.64T + 7T^{2} \) |
| 11 | \( 1 + 0.354T + 11T^{2} \) |
| 13 | \( 1 + 0.354T + 13T^{2} \) |
| 17 | \( 1 - 4T + 17T^{2} \) |
| 23 | \( 1 + 9.29T + 23T^{2} \) |
| 29 | \( 1 + 8.93T + 29T^{2} \) |
| 31 | \( 1 - 6T + 31T^{2} \) |
| 37 | \( 1 - 3.64T + 37T^{2} \) |
| 41 | \( 1 - 9.64T + 41T^{2} \) |
| 43 | \( 1 - 5.64T + 43T^{2} \) |
| 47 | \( 1 - 1.29T + 47T^{2} \) |
| 53 | \( 1 + 11.2T + 53T^{2} \) |
| 59 | \( 1 + 11.2T + 59T^{2} \) |
| 61 | \( 1 + 11.2T + 61T^{2} \) |
| 67 | \( 1 + 6.58T + 67T^{2} \) |
| 71 | \( 1 + 7.29T + 71T^{2} \) |
| 73 | \( 1 - 10T + 73T^{2} \) |
| 79 | \( 1 - 6.58T + 79T^{2} \) |
| 83 | \( 1 + 6T + 83T^{2} \) |
| 89 | \( 1 - 16.9T + 89T^{2} \) |
| 97 | \( 1 + 2.93T + 97T^{2} \) |
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
−10.67271511362287841677187630475, −9.516934784813309500688880154964, −7.80089205681565877940778521648, −7.71036418310382896784177414998, −6.27725230823798878770474023582, −5.71537394554387100462234331872, −4.64581092554730096159220447587, −4.02563579301204193992383587525, −2.98971781948011649321538550987, −1.79827696998905858967763090168,
1.79827696998905858967763090168, 2.98971781948011649321538550987, 4.02563579301204193992383587525, 4.64581092554730096159220447587, 5.71537394554387100462234331872, 6.27725230823798878770474023582, 7.71036418310382896784177414998, 7.80089205681565877940778521648, 9.516934784813309500688880154964, 10.67271511362287841677187630475