L(s) = 1 | − 8·2-s + 27·3-s + 64·4-s + 455.·5-s − 216·6-s − 343·7-s − 512·8-s + 729·9-s − 3.64e3·10-s − 7.83e3·11-s + 1.72e3·12-s − 2.19e3·13-s + 2.74e3·14-s + 1.23e4·15-s + 4.09e3·16-s − 2.74e4·17-s − 5.83e3·18-s + 4.35e4·19-s + 2.91e4·20-s − 9.26e3·21-s + 6.27e4·22-s − 7.13e4·23-s − 1.38e4·24-s + 1.29e5·25-s + 1.75e4·26-s + 1.96e4·27-s − 2.19e4·28-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 0.577·3-s + 0.5·4-s + 1.63·5-s − 0.408·6-s − 0.377·7-s − 0.353·8-s + 0.333·9-s − 1.15·10-s − 1.77·11-s + 0.288·12-s − 0.277·13-s + 0.267·14-s + 0.941·15-s + 0.250·16-s − 1.35·17-s − 0.235·18-s + 1.45·19-s + 0.815·20-s − 0.218·21-s + 1.25·22-s − 1.22·23-s − 0.204·24-s + 1.65·25-s + 0.196·26-s + 0.192·27-s − 0.188·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(4)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{9}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + 8T \) |
| 3 | \( 1 - 27T \) |
| 7 | \( 1 + 343T \) |
| 13 | \( 1 + 2.19e3T \) |
good | 5 | \( 1 - 455.T + 7.81e4T^{2} \) |
| 11 | \( 1 + 7.83e3T + 1.94e7T^{2} \) |
| 17 | \( 1 + 2.74e4T + 4.10e8T^{2} \) |
| 19 | \( 1 - 4.35e4T + 8.93e8T^{2} \) |
| 23 | \( 1 + 7.13e4T + 3.40e9T^{2} \) |
| 29 | \( 1 - 1.86e5T + 1.72e10T^{2} \) |
| 31 | \( 1 - 9.62e4T + 2.75e10T^{2} \) |
| 37 | \( 1 - 5.40e5T + 9.49e10T^{2} \) |
| 41 | \( 1 - 1.98e5T + 1.94e11T^{2} \) |
| 43 | \( 1 + 4.70e5T + 2.71e11T^{2} \) |
| 47 | \( 1 + 1.08e6T + 5.06e11T^{2} \) |
| 53 | \( 1 - 1.51e6T + 1.17e12T^{2} \) |
| 59 | \( 1 + 1.52e6T + 2.48e12T^{2} \) |
| 61 | \( 1 + 1.16e5T + 3.14e12T^{2} \) |
| 67 | \( 1 + 2.66e5T + 6.06e12T^{2} \) |
| 71 | \( 1 + 4.83e6T + 9.09e12T^{2} \) |
| 73 | \( 1 + 4.30e6T + 1.10e13T^{2} \) |
| 79 | \( 1 + 1.89e6T + 1.92e13T^{2} \) |
| 83 | \( 1 - 1.41e6T + 2.71e13T^{2} \) |
| 89 | \( 1 + 5.66e6T + 4.42e13T^{2} \) |
| 97 | \( 1 + 7.63e6T + 8.07e13T^{2} \) |
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
−9.392879449135923891677849728840, −8.440233227208890047657091051059, −7.61191871460841074015045570634, −6.54227806739356783945548897269, −5.73012324858555500752006371706, −4.70663289559954272601357708602, −2.79106971558309027499388656700, −2.49434092365327078206168752897, −1.35734377446350301625855024412, 0,
1.35734377446350301625855024412, 2.49434092365327078206168752897, 2.79106971558309027499388656700, 4.70663289559954272601357708602, 5.73012324858555500752006371706, 6.54227806739356783945548897269, 7.61191871460841074015045570634, 8.440233227208890047657091051059, 9.392879449135923891677849728840