| L(s) = 1 | − 13.8·2-s − 27·3-s + 63.2·4-s − 149.·5-s + 373.·6-s + 14.2·7-s + 895.·8-s + 729·9-s + 2.06e3·10-s + 531.·11-s − 1.70e3·12-s + 4.16e3·13-s − 197.·14-s + 4.03e3·15-s − 2.04e4·16-s + 4.56e3·17-s − 1.00e4·18-s + 3.18e4·19-s − 9.45e3·20-s − 385.·21-s − 7.35e3·22-s − 5.60e4·23-s − 2.41e4·24-s − 5.57e4·25-s − 5.76e4·26-s − 1.96e4·27-s + 903.·28-s + ⋯ |
| L(s) = 1 | − 1.22·2-s − 0.577·3-s + 0.494·4-s − 0.534·5-s + 0.705·6-s + 0.0157·7-s + 0.618·8-s + 0.333·9-s + 0.653·10-s + 0.120·11-s − 0.285·12-s + 0.526·13-s − 0.0192·14-s + 0.308·15-s − 1.24·16-s + 0.225·17-s − 0.407·18-s + 1.06·19-s − 0.264·20-s − 0.00908·21-s − 0.147·22-s − 0.960·23-s − 0.356·24-s − 0.714·25-s − 0.643·26-s − 0.192·27-s + 0.00777·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(0.5520223296\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5520223296\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + 27T \) |
| 59 | \( 1 + 2.05e5T \) |
| good | 2 | \( 1 + 13.8T + 128T^{2} \) |
| 5 | \( 1 + 149.T + 7.81e4T^{2} \) |
| 7 | \( 1 - 14.2T + 8.23e5T^{2} \) |
| 11 | \( 1 - 531.T + 1.94e7T^{2} \) |
| 13 | \( 1 - 4.16e3T + 6.27e7T^{2} \) |
| 17 | \( 1 - 4.56e3T + 4.10e8T^{2} \) |
| 19 | \( 1 - 3.18e4T + 8.93e8T^{2} \) |
| 23 | \( 1 + 5.60e4T + 3.40e9T^{2} \) |
| 29 | \( 1 - 5.20e4T + 1.72e10T^{2} \) |
| 31 | \( 1 + 4.40e4T + 2.75e10T^{2} \) |
| 37 | \( 1 - 1.13e4T + 9.49e10T^{2} \) |
| 41 | \( 1 + 4.55e5T + 1.94e11T^{2} \) |
| 43 | \( 1 - 1.77e5T + 2.71e11T^{2} \) |
| 47 | \( 1 + 1.27e6T + 5.06e11T^{2} \) |
| 53 | \( 1 - 2.74e5T + 1.17e12T^{2} \) |
| 61 | \( 1 + 1.52e5T + 3.14e12T^{2} \) |
| 67 | \( 1 - 2.06e6T + 6.06e12T^{2} \) |
| 71 | \( 1 + 1.79e6T + 9.09e12T^{2} \) |
| 73 | \( 1 - 3.32e6T + 1.10e13T^{2} \) |
| 79 | \( 1 - 6.43e6T + 1.92e13T^{2} \) |
| 83 | \( 1 + 1.15e6T + 2.71e13T^{2} \) |
| 89 | \( 1 - 3.69e6T + 4.42e13T^{2} \) |
| 97 | \( 1 - 6.33e6T + 8.07e13T^{2} \) |
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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
−11.28341085191120846664776020940, −10.23152997692028534914406648071, −9.482682211436727287208221709578, −8.286444338829260981001614593107, −7.57754871875625220506143784937, −6.39618270354167257928933158627, −4.98552690989814575959950772411, −3.66551492664650442096727457412, −1.67387161851102881113617326463, −0.52393267577116903636853752988,
0.52393267577116903636853752988, 1.67387161851102881113617326463, 3.66551492664650442096727457412, 4.98552690989814575959950772411, 6.39618270354167257928933158627, 7.57754871875625220506143784937, 8.286444338829260981001614593107, 9.482682211436727287208221709578, 10.23152997692028534914406648071, 11.28341085191120846664776020940
