| L(s) = 1 | + (−63 − 50.9i)3-s + 576. i·5-s − 2.78e3·7-s + (1.37e3 + 6.41e3i)9-s − 2.24e4i·11-s + 1.31e4·13-s + (2.93e4 − 3.63e4i)15-s + 6.63e4i·17-s + 1.44e5·19-s + (1.75e5 + 1.41e5i)21-s + 4.93e4i·23-s + 5.76e4·25-s + (2.39e5 − 4.74e5i)27-s − 6.27e5i·29-s − 7.28e5·31-s + ⋯ |
| L(s) = 1 | + (−0.777 − 0.628i)3-s + 0.923i·5-s − 1.16·7-s + (0.209 + 0.977i)9-s − 1.53i·11-s + 0.460·13-s + (0.580 − 0.718i)15-s + 0.794i·17-s + 1.10·19-s + (0.902 + 0.729i)21-s + 0.176i·23-s + 0.147·25-s + (0.451 − 0.892i)27-s − 0.887i·29-s − 0.789·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 192 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.777 - 0.628i)\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 192 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (-0.777 - 0.628i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{9}{2})\) |
\(\approx\) |
\(0.2488145699\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2488145699\) |
| \(L(5)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (63 + 50.9i)T \) |
| good | 5 | \( 1 - 576. iT - 3.90e5T^{2} \) |
| 7 | \( 1 + 2.78e3T + 5.76e6T^{2} \) |
| 11 | \( 1 + 2.24e4iT - 2.14e8T^{2} \) |
| 13 | \( 1 - 1.31e4T + 8.15e8T^{2} \) |
| 17 | \( 1 - 6.63e4iT - 6.97e9T^{2} \) |
| 19 | \( 1 - 1.44e5T + 1.69e10T^{2} \) |
| 23 | \( 1 - 4.93e4iT - 7.83e10T^{2} \) |
| 29 | \( 1 + 6.27e5iT - 5.00e11T^{2} \) |
| 31 | \( 1 + 7.28e5T + 8.52e11T^{2} \) |
| 37 | \( 1 - 1.96e6T + 3.51e12T^{2} \) |
| 41 | \( 1 + 9.86e5iT - 7.98e12T^{2} \) |
| 43 | \( 1 + 7.81e4T + 1.16e13T^{2} \) |
| 47 | \( 1 + 3.51e6iT - 2.38e13T^{2} \) |
| 53 | \( 1 - 5.22e5iT - 6.22e13T^{2} \) |
| 59 | \( 1 - 5.00e6iT - 1.46e14T^{2} \) |
| 61 | \( 1 + 1.75e7T + 1.91e14T^{2} \) |
| 67 | \( 1 + 1.71e7T + 4.06e14T^{2} \) |
| 71 | \( 1 - 2.58e7iT - 6.45e14T^{2} \) |
| 73 | \( 1 - 2.81e7T + 8.06e14T^{2} \) |
| 79 | \( 1 + 9.18e6T + 1.51e15T^{2} \) |
| 83 | \( 1 - 8.71e7iT - 2.25e15T^{2} \) |
| 89 | \( 1 - 8.12e7iT - 3.93e15T^{2} \) |
| 97 | \( 1 + 1.28e8T + 7.83e15T^{2} \) |
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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
−11.27842132424535676489518340011, −10.74702631597041894625329361422, −9.649564183896836964609948613275, −8.265564573597418229813744956856, −7.12331809745398978894000531445, −6.23277526712282718603787827700, −5.66459456218300965103038959200, −3.69446511946735511213971824553, −2.73354332234528429171769849027, −1.04908825119493175411588131662,
0.079939641677210007811450181877, 1.26385910026616628852121778392, 3.16310398887605375498984123926, 4.44488754505901311002922659413, 5.23191954487186675238405664344, 6.41154097679017247876098822429, 7.43297184495942713747825060255, 9.226629261037411717674908961276, 9.500713611456830895797617202516, 10.55797485071300290038199046690
