| L(s) = 1 | − 2.40·2-s − 27·3-s − 122.·4-s − 505.·5-s + 64.8·6-s + 1.40e3·7-s + 600.·8-s + 729·9-s + 1.21e3·10-s − 1.33e3·11-s + 3.30e3·12-s − 4.33e3·13-s − 3.36e3·14-s + 1.36e4·15-s + 1.42e4·16-s + 2.85e4·17-s − 1.75e3·18-s − 5.37e3·19-s + 6.18e4·20-s − 3.78e4·21-s + 3.19e3·22-s − 8.54e4·23-s − 1.62e4·24-s + 1.77e5·25-s + 1.04e4·26-s − 1.96e4·27-s − 1.71e5·28-s + ⋯ |
| L(s) = 1 | − 0.212·2-s − 0.577·3-s − 0.954·4-s − 1.80·5-s + 0.122·6-s + 1.54·7-s + 0.414·8-s + 0.333·9-s + 0.383·10-s − 0.301·11-s + 0.551·12-s − 0.547·13-s − 0.327·14-s + 1.04·15-s + 0.866·16-s + 1.40·17-s − 0.0707·18-s − 0.179·19-s + 1.72·20-s − 0.891·21-s + 0.0639·22-s − 1.46·23-s − 0.239·24-s + 2.27·25-s + 0.116·26-s − 0.192·27-s − 1.47·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 33 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 33 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(0.7175981286\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7175981286\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + 27T \) |
| 11 | \( 1 + 1.33e3T \) |
| good | 2 | \( 1 + 2.40T + 128T^{2} \) |
| 5 | \( 1 + 505.T + 7.81e4T^{2} \) |
| 7 | \( 1 - 1.40e3T + 8.23e5T^{2} \) |
| 13 | \( 1 + 4.33e3T + 6.27e7T^{2} \) |
| 17 | \( 1 - 2.85e4T + 4.10e8T^{2} \) |
| 19 | \( 1 + 5.37e3T + 8.93e8T^{2} \) |
| 23 | \( 1 + 8.54e4T + 3.40e9T^{2} \) |
| 29 | \( 1 - 5.94e3T + 1.72e10T^{2} \) |
| 31 | \( 1 - 2.84e5T + 2.75e10T^{2} \) |
| 37 | \( 1 - 9.82e4T + 9.49e10T^{2} \) |
| 41 | \( 1 - 9.60e4T + 1.94e11T^{2} \) |
| 43 | \( 1 + 1.98e5T + 2.71e11T^{2} \) |
| 47 | \( 1 - 2.17e5T + 5.06e11T^{2} \) |
| 53 | \( 1 - 8.44e5T + 1.17e12T^{2} \) |
| 59 | \( 1 + 1.14e6T + 2.48e12T^{2} \) |
| 61 | \( 1 - 3.20e6T + 3.14e12T^{2} \) |
| 67 | \( 1 - 2.09e6T + 6.06e12T^{2} \) |
| 71 | \( 1 - 6.41e5T + 9.09e12T^{2} \) |
| 73 | \( 1 + 3.25e6T + 1.10e13T^{2} \) |
| 79 | \( 1 - 7.68e6T + 1.92e13T^{2} \) |
| 83 | \( 1 - 4.06e6T + 2.71e13T^{2} \) |
| 89 | \( 1 - 1.64e6T + 4.42e13T^{2} \) |
| 97 | \( 1 + 3.61e6T + 8.07e13T^{2} \) |
| 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
−15.15135573466889985734975826668, −14.17720835345443643393402064150, −12.30487448005142951221979649038, −11.60983804246546302531416540358, −10.26430602955328430821602256398, −8.219846220628577470480357523172, −7.73795537709544526707996100357, −5.06677085602354374016002269664, −4.07821256261418294211626003997, −0.74013886104304199327858929722,
0.74013886104304199327858929722, 4.07821256261418294211626003997, 5.06677085602354374016002269664, 7.73795537709544526707996100357, 8.219846220628577470480357523172, 10.26430602955328430821602256398, 11.60983804246546302531416540358, 12.30487448005142951221979649038, 14.17720835345443643393402064150, 15.15135573466889985734975826668
