| L(s) = 1 | − 16.2·2-s + 136.·4-s + 125·5-s + 360.·7-s − 137.·8-s − 2.03e3·10-s − 3.88e3·11-s − 8.91e3·13-s − 5.86e3·14-s − 1.52e4·16-s − 2.15e4·17-s − 2.41e4·19-s + 1.70e4·20-s + 6.31e4·22-s − 6.91e4·23-s + 1.56e4·25-s + 1.44e5·26-s + 4.91e4·28-s + 6.60e4·29-s − 1.20e5·31-s + 2.65e5·32-s + 3.51e5·34-s + 4.50e4·35-s − 5.11e5·37-s + 3.92e5·38-s − 1.72e4·40-s − 8.87e4·41-s + ⋯ |
| L(s) = 1 | − 1.43·2-s + 1.06·4-s + 0.447·5-s + 0.397·7-s − 0.0952·8-s − 0.642·10-s − 0.879·11-s − 1.12·13-s − 0.570·14-s − 0.929·16-s − 1.06·17-s − 0.807·19-s + 0.476·20-s + 1.26·22-s − 1.18·23-s + 0.199·25-s + 1.61·26-s + 0.423·28-s + 0.502·29-s − 0.728·31-s + 1.43·32-s + 1.53·34-s + 0.177·35-s − 1.65·37-s + 1.16·38-s − 0.0425·40-s − 0.201·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 405 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 405 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(0.4309345509\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4309345509\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 - 125T \) |
| good | 2 | \( 1 + 16.2T + 128T^{2} \) |
| 7 | \( 1 - 360.T + 8.23e5T^{2} \) |
| 11 | \( 1 + 3.88e3T + 1.94e7T^{2} \) |
| 13 | \( 1 + 8.91e3T + 6.27e7T^{2} \) |
| 17 | \( 1 + 2.15e4T + 4.10e8T^{2} \) |
| 19 | \( 1 + 2.41e4T + 8.93e8T^{2} \) |
| 23 | \( 1 + 6.91e4T + 3.40e9T^{2} \) |
| 29 | \( 1 - 6.60e4T + 1.72e10T^{2} \) |
| 31 | \( 1 + 1.20e5T + 2.75e10T^{2} \) |
| 37 | \( 1 + 5.11e5T + 9.49e10T^{2} \) |
| 41 | \( 1 + 8.87e4T + 1.94e11T^{2} \) |
| 43 | \( 1 - 3.73e5T + 2.71e11T^{2} \) |
| 47 | \( 1 - 5.64e5T + 5.06e11T^{2} \) |
| 53 | \( 1 - 1.82e6T + 1.17e12T^{2} \) |
| 59 | \( 1 + 4.78e5T + 2.48e12T^{2} \) |
| 61 | \( 1 - 2.24e6T + 3.14e12T^{2} \) |
| 67 | \( 1 + 8.88e5T + 6.06e12T^{2} \) |
| 71 | \( 1 - 1.70e6T + 9.09e12T^{2} \) |
| 73 | \( 1 + 2.60e6T + 1.10e13T^{2} \) |
| 79 | \( 1 + 3.29e6T + 1.92e13T^{2} \) |
| 83 | \( 1 + 1.97e6T + 2.71e13T^{2} \) |
| 89 | \( 1 - 6.14e6T + 4.42e13T^{2} \) |
| 97 | \( 1 + 1.45e7T + 8.07e13T^{2} \) |
| 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
−10.16513347494308118355402719990, −9.088025295105432598004444314846, −8.414826174195539599096201783942, −7.52706440163001187710003856635, −6.71518654960724580003811856854, −5.38238798614537616326486621896, −4.33477533700773220745590358670, −2.43653735029493671510097423060, −1.85097534499677501947784441850, −0.35919885920128324692173089109,
0.35919885920128324692173089109, 1.85097534499677501947784441850, 2.43653735029493671510097423060, 4.33477533700773220745590358670, 5.38238798614537616326486621896, 6.71518654960724580003811856854, 7.52706440163001187710003856635, 8.414826174195539599096201783942, 9.088025295105432598004444314846, 10.16513347494308118355402719990
