| L(s) = 1 | − 15.4·2-s + 32.1·3-s + 111.·4-s − 143.·5-s − 497.·6-s + 558.·7-s + 259.·8-s − 1.15e3·9-s + 2.21e3·10-s + 3.58e3·12-s − 842.·13-s − 8.63e3·14-s − 4.60e3·15-s − 1.82e4·16-s + 8.60e3·17-s + 1.77e4·18-s + 3.92e4·19-s − 1.59e4·20-s + 1.79e4·21-s + 6.03e4·23-s + 8.35e3·24-s − 5.76e4·25-s + 1.30e4·26-s − 1.07e5·27-s + 6.21e4·28-s − 2.05e5·29-s + 7.12e4·30-s + ⋯ |
| L(s) = 1 | − 1.36·2-s + 0.688·3-s + 0.868·4-s − 0.511·5-s − 0.941·6-s + 0.615·7-s + 0.179·8-s − 0.526·9-s + 0.699·10-s + 0.598·12-s − 0.106·13-s − 0.841·14-s − 0.352·15-s − 1.11·16-s + 0.424·17-s + 0.719·18-s + 1.31·19-s − 0.444·20-s + 0.423·21-s + 1.03·23-s + 0.123·24-s − 0.738·25-s + 0.145·26-s − 1.05·27-s + 0.534·28-s − 1.56·29-s + 0.481·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 121 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 121 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(1.035653983\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.035653983\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 11 | \( 1 \) |
| good | 2 | \( 1 + 15.4T + 128T^{2} \) |
| 3 | \( 1 - 32.1T + 2.18e3T^{2} \) |
| 5 | \( 1 + 143.T + 7.81e4T^{2} \) |
| 7 | \( 1 - 558.T + 8.23e5T^{2} \) |
| 13 | \( 1 + 842.T + 6.27e7T^{2} \) |
| 17 | \( 1 - 8.60e3T + 4.10e8T^{2} \) |
| 19 | \( 1 - 3.92e4T + 8.93e8T^{2} \) |
| 23 | \( 1 - 6.03e4T + 3.40e9T^{2} \) |
| 29 | \( 1 + 2.05e5T + 1.72e10T^{2} \) |
| 31 | \( 1 + 2.74e5T + 2.75e10T^{2} \) |
| 37 | \( 1 - 3.66e5T + 9.49e10T^{2} \) |
| 41 | \( 1 - 1.54e5T + 1.94e11T^{2} \) |
| 43 | \( 1 - 2.33e5T + 2.71e11T^{2} \) |
| 47 | \( 1 - 5.50e5T + 5.06e11T^{2} \) |
| 53 | \( 1 - 1.29e6T + 1.17e12T^{2} \) |
| 59 | \( 1 - 8.08e5T + 2.48e12T^{2} \) |
| 61 | \( 1 - 2.04e6T + 3.14e12T^{2} \) |
| 67 | \( 1 + 5.07e5T + 6.06e12T^{2} \) |
| 71 | \( 1 - 3.33e6T + 9.09e12T^{2} \) |
| 73 | \( 1 - 2.46e6T + 1.10e13T^{2} \) |
| 79 | \( 1 + 4.29e5T + 1.92e13T^{2} \) |
| 83 | \( 1 - 9.75e6T + 2.71e13T^{2} \) |
| 89 | \( 1 - 3.43e6T + 4.42e13T^{2} \) |
| 97 | \( 1 - 1.18e7T + 8.07e13T^{2} \) |
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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
−11.59882237958530396006842806401, −11.00516215091616270744132430619, −9.584777254407689201549706152287, −8.958636511822057210198650718470, −7.84039942196367921186328814966, −7.42156616245120014257381352514, −5.38761811719279805580132161054, −3.66966275953755361020428938657, −2.12486540625712302172367458622, −0.72408610915640228370480414712,
0.72408610915640228370480414712, 2.12486540625712302172367458622, 3.66966275953755361020428938657, 5.38761811719279805580132161054, 7.42156616245120014257381352514, 7.84039942196367921186328814966, 8.958636511822057210198650718470, 9.584777254407689201549706152287, 11.00516215091616270744132430619, 11.59882237958530396006842806401
