| L(s) = 1 | + 89.5·2-s − 243·3-s + 5.97e3·4-s − 2.39e3·5-s − 2.17e4·6-s + 8.27e4·7-s + 3.51e5·8-s + 5.90e4·9-s − 2.14e5·10-s + 7.79e5·11-s − 1.45e6·12-s − 3.67e5·13-s + 7.41e6·14-s + 5.81e5·15-s + 1.92e7·16-s + 3.35e6·17-s + 5.28e6·18-s − 1.92e6·19-s − 1.43e7·20-s − 2.01e7·21-s + 6.98e7·22-s − 3.41e7·23-s − 8.54e7·24-s − 4.30e7·25-s − 3.29e7·26-s − 1.43e7·27-s + 4.94e8·28-s + ⋯ |
| L(s) = 1 | + 1.97·2-s − 0.577·3-s + 2.91·4-s − 0.342·5-s − 1.14·6-s + 1.86·7-s + 3.79·8-s + 0.333·9-s − 0.678·10-s + 1.45·11-s − 1.68·12-s − 0.274·13-s + 3.68·14-s + 0.197·15-s + 4.59·16-s + 0.573·17-s + 0.659·18-s − 0.178·19-s − 0.999·20-s − 1.07·21-s + 2.88·22-s − 1.10·23-s − 2.19·24-s − 0.882·25-s − 0.543·26-s − 0.192·27-s + 5.43·28-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
| \(L(6)\) |
\(\approx\) |
\(10.44013098\) |
| \(L(\frac12)\) |
\(\approx\) |
\(10.44013098\) |
| \(L(\frac{13}{2})\) |
|
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 | \( 1 - 89.5T + 2.04e3T^{2} \) |
| 5 | \( 1 + 2.39e3T + 4.88e7T^{2} \) |
| 7 | \( 1 - 8.27e4T + 1.97e9T^{2} \) |
| 11 | \( 1 - 7.79e5T + 2.85e11T^{2} \) |
| 13 | \( 1 + 3.67e5T + 1.79e12T^{2} \) |
| 17 | \( 1 - 3.35e6T + 3.42e13T^{2} \) |
| 19 | \( 1 + 1.92e6T + 1.16e14T^{2} \) |
| 23 | \( 1 + 3.41e7T + 9.52e14T^{2} \) |
| 29 | \( 1 + 2.65e7T + 1.22e16T^{2} \) |
| 31 | \( 1 - 1.71e8T + 2.54e16T^{2} \) |
| 37 | \( 1 + 7.07e8T + 1.77e17T^{2} \) |
| 41 | \( 1 + 2.76e8T + 5.50e17T^{2} \) |
| 43 | \( 1 + 9.77e8T + 9.29e17T^{2} \) |
| 47 | \( 1 - 2.45e9T + 2.47e18T^{2} \) |
| 53 | \( 1 + 3.75e9T + 9.26e18T^{2} \) |
| 61 | \( 1 - 1.05e9T + 4.35e19T^{2} \) |
| 67 | \( 1 + 4.59e9T + 1.22e20T^{2} \) |
| 71 | \( 1 - 1.72e10T + 2.31e20T^{2} \) |
| 73 | \( 1 - 2.89e9T + 3.13e20T^{2} \) |
| 79 | \( 1 + 5.08e10T + 7.47e20T^{2} \) |
| 83 | \( 1 - 3.23e10T + 1.28e21T^{2} \) |
| 89 | \( 1 - 1.57e10T + 2.77e21T^{2} \) |
| 97 | \( 1 - 1.26e11T + 7.15e21T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−11.30088083206591501250457118190, −10.29690066619403446614261888717, −8.145569371057700546869829686759, −7.23056595479817638434278616280, −6.16783470192277308214757963834, −5.21882102297128147312225518168, −4.41355136098144941503621250830, −3.69903970037825725108062669318, −2.00240563608223030868577907964, −1.31289105435087891076286762035,
1.31289105435087891076286762035, 2.00240563608223030868577907964, 3.69903970037825725108062669318, 4.41355136098144941503621250830, 5.21882102297128147312225518168, 6.16783470192277308214757963834, 7.23056595479817638434278616280, 8.145569371057700546869829686759, 10.29690066619403446614261888717, 11.30088083206591501250457118190
