L(s) = 1 | − 2-s + 4-s + 5-s − 7-s − 8-s − 10-s + 6.12·11-s + 5.68·13-s + 14-s + 16-s + 5·17-s + 0.561·19-s + 20-s − 6.12·22-s + 6.56·23-s + 25-s − 5.68·26-s − 28-s + 6.68·29-s − 3.56·31-s − 32-s − 5·34-s − 35-s + 37-s − 0.561·38-s − 40-s − 8.68·41-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 0.5·4-s + 0.447·5-s − 0.377·7-s − 0.353·8-s − 0.316·10-s + 1.84·11-s + 1.57·13-s + 0.267·14-s + 0.250·16-s + 1.21·17-s + 0.128·19-s + 0.223·20-s − 1.30·22-s + 1.36·23-s + 0.200·25-s − 1.11·26-s − 0.188·28-s + 1.24·29-s − 0.639·31-s − 0.176·32-s − 0.857·34-s − 0.169·35-s + 0.164·37-s − 0.0910·38-s − 0.158·40-s − 1.35·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3330 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3330 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.916393557\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.916393557\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 37 | \( 1 - T \) |
good | 7 | \( 1 + T + 7T^{2} \) |
| 11 | \( 1 - 6.12T + 11T^{2} \) |
| 13 | \( 1 - 5.68T + 13T^{2} \) |
| 17 | \( 1 - 5T + 17T^{2} \) |
| 19 | \( 1 - 0.561T + 19T^{2} \) |
| 23 | \( 1 - 6.56T + 23T^{2} \) |
| 29 | \( 1 - 6.68T + 29T^{2} \) |
| 31 | \( 1 + 3.56T + 31T^{2} \) |
| 41 | \( 1 + 8.68T + 41T^{2} \) |
| 43 | \( 1 + 12.6T + 43T^{2} \) |
| 47 | \( 1 - 8T + 47T^{2} \) |
| 53 | \( 1 + 5.24T + 53T^{2} \) |
| 59 | \( 1 + 4.24T + 59T^{2} \) |
| 61 | \( 1 + 13.8T + 61T^{2} \) |
| 67 | \( 1 - 13.1T + 67T^{2} \) |
| 71 | \( 1 + 1.12T + 71T^{2} \) |
| 73 | \( 1 + 6.56T + 73T^{2} \) |
| 79 | \( 1 - 14.2T + 79T^{2} \) |
| 83 | \( 1 - 7.43T + 83T^{2} \) |
| 89 | \( 1 + 5.68T + 89T^{2} \) |
| 97 | \( 1 - 3.31T + 97T^{2} \) |
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
−8.738096859714066783884858157454, −8.108745067018923037665946059083, −6.96334737231876774116240766277, −6.51468381983738416972402312997, −5.91088794918801038622547048035, −4.86767174240516691556533714056, −3.59758067043958126172713071718, −3.18545645911725068986873810059, −1.58308490351461272300127865148, −1.06501804660938380048832710219,
1.06501804660938380048832710219, 1.58308490351461272300127865148, 3.18545645911725068986873810059, 3.59758067043958126172713071718, 4.86767174240516691556533714056, 5.91088794918801038622547048035, 6.51468381983738416972402312997, 6.96334737231876774116240766277, 8.108745067018923037665946059083, 8.738096859714066783884858157454