L(s) = 1 | + 43.4·2-s − 243·3-s − 156.·4-s + 1.83e3·5-s − 1.05e4·6-s + 3.38e4·7-s − 9.58e4·8-s + 5.90e4·9-s + 7.97e4·10-s + 9.08e5·11-s + 3.81e4·12-s + 5.03e5·13-s + 1.47e6·14-s − 4.45e5·15-s − 3.84e6·16-s + 4.40e6·17-s + 2.56e6·18-s + 1.49e6·19-s − 2.87e5·20-s − 8.22e6·21-s + 3.94e7·22-s − 1.40e7·23-s + 2.33e7·24-s − 4.54e7·25-s + 2.18e7·26-s − 1.43e7·27-s − 5.31e6·28-s + ⋯ |
L(s) = 1 | + 0.960·2-s − 0.577·3-s − 0.0766·4-s + 0.262·5-s − 0.554·6-s + 0.761·7-s − 1.03·8-s + 0.333·9-s + 0.252·10-s + 1.70·11-s + 0.0442·12-s + 0.376·13-s + 0.731·14-s − 0.151·15-s − 0.917·16-s + 0.752·17-s + 0.320·18-s + 0.138·19-s − 0.0201·20-s − 0.439·21-s + 1.63·22-s − 0.455·23-s + 0.597·24-s − 0.931·25-s + 0.361·26-s − 0.192·27-s − 0.0583·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(12-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s+11/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(6)\) |
\(\approx\) |
\(3.419474226\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.419474226\) |
\(L(\frac{13}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + 243T \) |
| 59 | \( 1 + 7.14e8T \) |
good | 2 | \( 1 - 43.4T + 2.04e3T^{2} \) |
| 5 | \( 1 - 1.83e3T + 4.88e7T^{2} \) |
| 7 | \( 1 - 3.38e4T + 1.97e9T^{2} \) |
| 11 | \( 1 - 9.08e5T + 2.85e11T^{2} \) |
| 13 | \( 1 - 5.03e5T + 1.79e12T^{2} \) |
| 17 | \( 1 - 4.40e6T + 3.42e13T^{2} \) |
| 19 | \( 1 - 1.49e6T + 1.16e14T^{2} \) |
| 23 | \( 1 + 1.40e7T + 9.52e14T^{2} \) |
| 29 | \( 1 + 7.57e7T + 1.22e16T^{2} \) |
| 31 | \( 1 + 1.41e8T + 2.54e16T^{2} \) |
| 37 | \( 1 + 4.72e8T + 1.77e17T^{2} \) |
| 41 | \( 1 - 1.06e9T + 5.50e17T^{2} \) |
| 43 | \( 1 - 1.00e9T + 9.29e17T^{2} \) |
| 47 | \( 1 - 1.98e9T + 2.47e18T^{2} \) |
| 53 | \( 1 - 4.29e9T + 9.26e18T^{2} \) |
| 61 | \( 1 - 6.72e8T + 4.35e19T^{2} \) |
| 67 | \( 1 + 5.16e9T + 1.22e20T^{2} \) |
| 71 | \( 1 + 1.10e10T + 2.31e20T^{2} \) |
| 73 | \( 1 - 1.07e10T + 3.13e20T^{2} \) |
| 79 | \( 1 - 5.11e10T + 7.47e20T^{2} \) |
| 83 | \( 1 + 6.20e10T + 1.28e21T^{2} \) |
| 89 | \( 1 - 4.27e10T + 2.77e21T^{2} \) |
| 97 | \( 1 - 1.93e10T + 7.15e21T^{2} \) |
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
−11.00661664668225732934010619782, −9.625445056971150776579579234192, −8.806840565782343628948452417088, −7.36938243520911524141018508899, −6.08237349527034369145874740892, −5.51575224484254580566684024840, −4.28283954633977940878200457456, −3.64128353811300947691458571809, −1.88654040157261111224040077174, −0.78066881283115041094195855392,
0.78066881283115041094195855392, 1.88654040157261111224040077174, 3.64128353811300947691458571809, 4.28283954633977940878200457456, 5.51575224484254580566684024840, 6.08237349527034369145874740892, 7.36938243520911524141018508899, 8.806840565782343628948452417088, 9.625445056971150776579579234192, 11.00661664668225732934010619782