| L(s) = 1 | − 10.3·2-s − 27·3-s − 20.4·4-s − 142.·5-s + 280.·6-s − 343·7-s + 1.53e3·8-s + 729·9-s + 1.48e3·10-s + 5.64e3·11-s + 552.·12-s + 4.17e3·13-s + 3.55e3·14-s + 3.86e3·15-s − 1.33e4·16-s + 7.76e3·17-s − 7.56e3·18-s − 3.09e4·19-s + 2.92e3·20-s + 9.26e3·21-s − 5.85e4·22-s + 3.37e4·23-s − 4.15e4·24-s − 5.76e4·25-s − 4.32e4·26-s − 1.96e4·27-s + 7.01e3·28-s + ⋯ |
| L(s) = 1 | − 0.916·2-s − 0.577·3-s − 0.159·4-s − 0.511·5-s + 0.529·6-s − 0.377·7-s + 1.06·8-s + 0.333·9-s + 0.468·10-s + 1.27·11-s + 0.0922·12-s + 0.526·13-s + 0.346·14-s + 0.295·15-s − 0.814·16-s + 0.383·17-s − 0.305·18-s − 1.03·19-s + 0.0817·20-s + 0.218·21-s − 1.17·22-s + 0.578·23-s − 0.613·24-s − 0.738·25-s − 0.482·26-s − 0.192·27-s + 0.0603·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 21 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 21 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(0.6536641283\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6536641283\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + 27T \) |
| 7 | \( 1 + 343T \) |
| good | 2 | \( 1 + 10.3T + 128T^{2} \) |
| 5 | \( 1 + 142.T + 7.81e4T^{2} \) |
| 11 | \( 1 - 5.64e3T + 1.94e7T^{2} \) |
| 13 | \( 1 - 4.17e3T + 6.27e7T^{2} \) |
| 17 | \( 1 - 7.76e3T + 4.10e8T^{2} \) |
| 19 | \( 1 + 3.09e4T + 8.93e8T^{2} \) |
| 23 | \( 1 - 3.37e4T + 3.40e9T^{2} \) |
| 29 | \( 1 - 2.04e5T + 1.72e10T^{2} \) |
| 31 | \( 1 - 2.79e5T + 2.75e10T^{2} \) |
| 37 | \( 1 + 4.72e4T + 9.49e10T^{2} \) |
| 41 | \( 1 - 4.88e5T + 1.94e11T^{2} \) |
| 43 | \( 1 - 5.31e5T + 2.71e11T^{2} \) |
| 47 | \( 1 + 2.47e5T + 5.06e11T^{2} \) |
| 53 | \( 1 + 6.51e5T + 1.17e12T^{2} \) |
| 59 | \( 1 + 1.47e5T + 2.48e12T^{2} \) |
| 61 | \( 1 + 1.64e6T + 3.14e12T^{2} \) |
| 67 | \( 1 - 4.25e6T + 6.06e12T^{2} \) |
| 71 | \( 1 - 3.23e6T + 9.09e12T^{2} \) |
| 73 | \( 1 - 2.53e6T + 1.10e13T^{2} \) |
| 79 | \( 1 - 5.72e6T + 1.92e13T^{2} \) |
| 83 | \( 1 + 6.06e6T + 2.71e13T^{2} \) |
| 89 | \( 1 + 2.65e6T + 4.42e13T^{2} \) |
| 97 | \( 1 + 1.53e7T + 8.07e13T^{2} \) |
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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
−16.88224499247058652691081534702, −15.70250906837755466712199071171, −13.95625076943507760994215666074, −12.35402637275694830083805626076, −10.97429203686337127180376045665, −9.613869564836525851004311774933, −8.264832141994961435531319042294, −6.54530664648629794277838584581, −4.24535835091368686150635760271, −0.885108843589874238964748372576,
0.885108843589874238964748372576, 4.24535835091368686150635760271, 6.54530664648629794277838584581, 8.264832141994961435531319042294, 9.613869564836525851004311774933, 10.97429203686337127180376045665, 12.35402637275694830083805626076, 13.95625076943507760994215666074, 15.70250906837755466712199071171, 16.88224499247058652691081534702
