| L(s) = 1 | − 72.0·2-s − 243·3-s + 3.13e3·4-s + 1.03e4·5-s + 1.75e4·6-s + 1.68e4·7-s − 7.85e4·8-s + 5.90e4·9-s − 7.48e5·10-s − 5.48e4·11-s − 7.62e5·12-s − 3.71e5·13-s − 1.21e6·14-s − 2.52e6·15-s − 7.70e5·16-s + 8.38e6·17-s − 4.25e6·18-s + 2.28e6·19-s + 3.26e7·20-s − 4.08e6·21-s + 3.95e6·22-s + 3.46e7·23-s + 1.90e7·24-s + 5.90e7·25-s + 2.67e7·26-s − 1.43e7·27-s + 5.27e7·28-s + ⋯ |
| L(s) = 1 | − 1.59·2-s − 0.577·3-s + 1.53·4-s + 1.48·5-s + 0.918·6-s + 0.377·7-s − 0.847·8-s + 0.333·9-s − 2.36·10-s − 0.102·11-s − 0.884·12-s − 0.277·13-s − 0.601·14-s − 0.858·15-s − 0.183·16-s + 1.43·17-s − 0.530·18-s + 0.211·19-s + 2.27·20-s − 0.218·21-s + 0.163·22-s + 1.12·23-s + 0.489·24-s + 1.20·25-s + 0.441·26-s − 0.192·27-s + 0.579·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(12-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s+11/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(6)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{13}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + 243T \) |
| 7 | \( 1 - 1.68e4T \) |
| 13 | \( 1 + 3.71e5T \) |
| good | 2 | \( 1 + 72.0T + 2.04e3T^{2} \) |
| 5 | \( 1 - 1.03e4T + 4.88e7T^{2} \) |
| 11 | \( 1 + 5.48e4T + 2.85e11T^{2} \) |
| 17 | \( 1 - 8.38e6T + 3.42e13T^{2} \) |
| 19 | \( 1 - 2.28e6T + 1.16e14T^{2} \) |
| 23 | \( 1 - 3.46e7T + 9.52e14T^{2} \) |
| 29 | \( 1 - 4.87e7T + 1.22e16T^{2} \) |
| 31 | \( 1 + 1.98e8T + 2.54e16T^{2} \) |
| 37 | \( 1 + 2.17e8T + 1.77e17T^{2} \) |
| 41 | \( 1 + 5.94e8T + 5.50e17T^{2} \) |
| 43 | \( 1 + 9.67e7T + 9.29e17T^{2} \) |
| 47 | \( 1 + 1.28e9T + 2.47e18T^{2} \) |
| 53 | \( 1 + 2.30e9T + 9.26e18T^{2} \) |
| 59 | \( 1 + 3.33e9T + 3.01e19T^{2} \) |
| 61 | \( 1 + 5.51e8T + 4.35e19T^{2} \) |
| 67 | \( 1 + 1.91e10T + 1.22e20T^{2} \) |
| 71 | \( 1 + 2.26e10T + 2.31e20T^{2} \) |
| 73 | \( 1 - 1.35e10T + 3.13e20T^{2} \) |
| 79 | \( 1 + 1.95e10T + 7.47e20T^{2} \) |
| 83 | \( 1 + 2.23e10T + 1.28e21T^{2} \) |
| 89 | \( 1 + 8.54e9T + 2.77e21T^{2} \) |
| 97 | \( 1 + 1.38e11T + 7.15e21T^{2} \) |
| 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
−9.611262901989018716497241641392, −8.823855681516873655253335949615, −7.68197010526159972001753078692, −6.83032217507165380264257094741, −5.77575414943583715589558102891, −4.95502942754692623322359726685, −2.94002481342536784414820767080, −1.67225266177460485655011513436, −1.24544456092908076262490881949, 0,
1.24544456092908076262490881949, 1.67225266177460485655011513436, 2.94002481342536784414820767080, 4.95502942754692623322359726685, 5.77575414943583715589558102891, 6.83032217507165380264257094741, 7.68197010526159972001753078692, 8.823855681516873655253335949615, 9.611262901989018716497241641392
