| L(s) = 1 | + 77.3·2-s + 243·3-s + 3.94e3·4-s + 1.24e4·5-s + 1.88e4·6-s + 7.72e4·7-s + 1.46e5·8-s + 5.90e4·9-s + 9.60e5·10-s − 7.26e5·11-s + 9.57e5·12-s + 5.02e5·13-s + 5.97e6·14-s + 3.01e6·15-s + 3.27e6·16-s + 5.93e6·17-s + 4.57e6·18-s − 2.85e5·19-s + 4.89e7·20-s + 1.87e7·21-s − 5.62e7·22-s − 3.99e7·23-s + 3.56e7·24-s + 1.05e8·25-s + 3.89e7·26-s + 1.43e7·27-s + 3.04e8·28-s + ⋯ |
| L(s) = 1 | + 1.71·2-s + 0.577·3-s + 1.92·4-s + 1.77·5-s + 0.987·6-s + 1.73·7-s + 1.58·8-s + 0.333·9-s + 3.03·10-s − 1.36·11-s + 1.11·12-s + 0.375·13-s + 2.97·14-s + 1.02·15-s + 0.780·16-s + 1.01·17-s + 0.570·18-s − 0.0264·19-s + 3.41·20-s + 1.00·21-s − 2.32·22-s − 1.29·23-s + 0.913·24-s + 2.15·25-s + 0.642·26-s + 0.192·27-s + 3.34·28-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
| \(L(6)\) |
\(\approx\) |
\(13.80081199\) |
| \(L(\frac12)\) |
\(\approx\) |
\(13.80081199\) |
| \(L(\frac{13}{2})\) |
|
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 | \( 1 - 77.3T + 2.04e3T^{2} \) |
| 5 | \( 1 - 1.24e4T + 4.88e7T^{2} \) |
| 7 | \( 1 - 7.72e4T + 1.97e9T^{2} \) |
| 11 | \( 1 + 7.26e5T + 2.85e11T^{2} \) |
| 13 | \( 1 - 5.02e5T + 1.79e12T^{2} \) |
| 17 | \( 1 - 5.93e6T + 3.42e13T^{2} \) |
| 19 | \( 1 + 2.85e5T + 1.16e14T^{2} \) |
| 23 | \( 1 + 3.99e7T + 9.52e14T^{2} \) |
| 29 | \( 1 + 1.60e8T + 1.22e16T^{2} \) |
| 31 | \( 1 - 1.72e8T + 2.54e16T^{2} \) |
| 37 | \( 1 + 4.59e8T + 1.77e17T^{2} \) |
| 41 | \( 1 + 5.35e7T + 5.50e17T^{2} \) |
| 43 | \( 1 + 7.31e8T + 9.29e17T^{2} \) |
| 47 | \( 1 + 7.00e8T + 2.47e18T^{2} \) |
| 53 | \( 1 + 5.47e8T + 9.26e18T^{2} \) |
| 61 | \( 1 + 6.37e9T + 4.35e19T^{2} \) |
| 67 | \( 1 + 8.96e9T + 1.22e20T^{2} \) |
| 71 | \( 1 - 6.32e9T + 2.31e20T^{2} \) |
| 73 | \( 1 + 1.17e10T + 3.13e20T^{2} \) |
| 79 | \( 1 + 2.87e10T + 7.47e20T^{2} \) |
| 83 | \( 1 + 5.71e10T + 1.28e21T^{2} \) |
| 89 | \( 1 - 7.05e10T + 2.77e21T^{2} \) |
| 97 | \( 1 + 1.00e11T + 7.15e21T^{2} \) |
| 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
−10.77249825737934982103060983639, −9.987484015381089462468391464945, −8.458599951029273231409470912959, −7.44059679185768842114303320471, −5.94704074827476693385348500903, −5.37466189622509153596839284361, −4.58945246394126700432869334870, −3.12802284963457442896053062608, −2.07114245643188834475573261288, −1.62221388012517821857623606530,
1.62221388012517821857623606530, 2.07114245643188834475573261288, 3.12802284963457442896053062608, 4.58945246394126700432869334870, 5.37466189622509153596839284361, 5.94704074827476693385348500903, 7.44059679185768842114303320471, 8.458599951029273231409470912959, 9.987484015381089462468391464945, 10.77249825737934982103060983639
