| L(s) = 1 | − 3-s + 2·5-s + 9-s + 11-s + 13-s − 2·15-s − 5.40·17-s − 7.40·19-s − 3.40·23-s − 25-s − 27-s + 2·29-s + 7.40·31-s − 33-s − 5.40·37-s − 39-s + 2·41-s + 2·45-s + 8·47-s − 7·49-s + 5.40·51-s + 9.40·53-s + 2·55-s + 7.40·57-s + 4·59-s + 12.8·61-s + 2·65-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.894·5-s + 0.333·9-s + 0.301·11-s + 0.277·13-s − 0.516·15-s − 1.31·17-s − 1.69·19-s − 0.709·23-s − 0.200·25-s − 0.192·27-s + 0.371·29-s + 1.32·31-s − 0.174·33-s − 0.888·37-s − 0.160·39-s + 0.312·41-s + 0.298·45-s + 1.16·47-s − 49-s + 0.756·51-s + 1.29·53-s + 0.269·55-s + 0.980·57-s + 0.520·59-s + 1.63·61-s + 0.248·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6864 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6864 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.671519907\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.671519907\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 11 | \( 1 - T \) |
| 13 | \( 1 - T \) |
| good | 5 | \( 1 - 2T + 5T^{2} \) |
| 7 | \( 1 + 7T^{2} \) |
| 17 | \( 1 + 5.40T + 17T^{2} \) |
| 19 | \( 1 + 7.40T + 19T^{2} \) |
| 23 | \( 1 + 3.40T + 23T^{2} \) |
| 29 | \( 1 - 2T + 29T^{2} \) |
| 31 | \( 1 - 7.40T + 31T^{2} \) |
| 37 | \( 1 + 5.40T + 37T^{2} \) |
| 41 | \( 1 - 2T + 41T^{2} \) |
| 43 | \( 1 + 43T^{2} \) |
| 47 | \( 1 - 8T + 47T^{2} \) |
| 53 | \( 1 - 9.40T + 53T^{2} \) |
| 59 | \( 1 - 4T + 59T^{2} \) |
| 61 | \( 1 - 12.8T + 61T^{2} \) |
| 67 | \( 1 + 67T^{2} \) |
| 71 | \( 1 - 14.8T + 71T^{2} \) |
| 73 | \( 1 - 13.4T + 73T^{2} \) |
| 79 | \( 1 - 10.8T + 79T^{2} \) |
| 83 | \( 1 - 4T + 83T^{2} \) |
| 89 | \( 1 + 10T + 89T^{2} \) |
| 97 | \( 1 - 10T + 97T^{2} \) |
| 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
−8.171816111586762181299312445602, −6.86186049447615591190331626880, −6.56955933708826323842523617123, −5.96615342131532729597835938087, −5.21498858539524618202167655528, −4.37818302280689094941677134715, −3.81465625105643884975462559220, −2.37665934569262341474386175164, −1.97103896557122132838201997330, −0.66679153370731697142654697181,
0.66679153370731697142654697181, 1.97103896557122132838201997330, 2.37665934569262341474386175164, 3.81465625105643884975462559220, 4.37818302280689094941677134715, 5.21498858539524618202167655528, 5.96615342131532729597835938087, 6.56955933708826323842523617123, 6.86186049447615591190331626880, 8.171816111586762181299312445602
