| L(s) = 1 | + 46.7·3-s + 721. i·5-s − 1.93e3i·7-s + 2.18e3·9-s − 1.44e4·11-s − 2.28e3i·13-s + 3.37e4i·15-s − 7.80e4·17-s + 4.84e4·19-s − 9.05e4i·21-s − 2.22e5i·23-s − 1.29e5·25-s + 1.02e5·27-s + 1.12e6i·29-s − 8.28e5i·31-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 1.15i·5-s − 0.806i·7-s + 0.333·9-s − 0.984·11-s − 0.0798i·13-s + 0.666i·15-s − 0.934·17-s + 0.371·19-s − 0.465i·21-s − 0.796i·23-s − 0.332·25-s + 0.192·27-s + 1.59i·29-s − 0.897i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & \, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{9}{2})\) |
\(\approx\) |
\(2.357795950\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.357795950\) |
| \(L(5)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - 46.7T \) |
| good | 5 | \( 1 - 721. iT - 3.90e5T^{2} \) |
| 7 | \( 1 + 1.93e3iT - 5.76e6T^{2} \) |
| 11 | \( 1 + 1.44e4T + 2.14e8T^{2} \) |
| 13 | \( 1 + 2.28e3iT - 8.15e8T^{2} \) |
| 17 | \( 1 + 7.80e4T + 6.97e9T^{2} \) |
| 19 | \( 1 - 4.84e4T + 1.69e10T^{2} \) |
| 23 | \( 1 + 2.22e5iT - 7.83e10T^{2} \) |
| 29 | \( 1 - 1.12e6iT - 5.00e11T^{2} \) |
| 31 | \( 1 + 8.28e5iT - 8.52e11T^{2} \) |
| 37 | \( 1 - 9.31e5iT - 3.51e12T^{2} \) |
| 41 | \( 1 - 4.58e6T + 7.98e12T^{2} \) |
| 43 | \( 1 + 2.91e6T + 1.16e13T^{2} \) |
| 47 | \( 1 + 7.52e6iT - 2.38e13T^{2} \) |
| 53 | \( 1 + 4.79e6iT - 6.22e13T^{2} \) |
| 59 | \( 1 - 8.36e6T + 1.46e14T^{2} \) |
| 61 | \( 1 + 3.86e6iT - 1.91e14T^{2} \) |
| 67 | \( 1 - 3.52e5T + 4.06e14T^{2} \) |
| 71 | \( 1 - 2.68e7iT - 6.45e14T^{2} \) |
| 73 | \( 1 - 8.13e6T + 8.06e14T^{2} \) |
| 79 | \( 1 + 4.31e7iT - 1.51e15T^{2} \) |
| 83 | \( 1 - 3.58e7T + 2.25e15T^{2} \) |
| 89 | \( 1 - 2.79e7T + 3.93e15T^{2} \) |
| 97 | \( 1 - 1.07e8T + 7.83e15T^{2} \) |
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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
−10.28690462896799943394518545311, −9.053217417426637146396883873941, −8.001660060664527422489226156640, −7.18568506437436125803940497971, −6.52789057048455450935720220473, −5.08959808761105153387096438054, −3.91373194707022530086224350439, −2.95991202800271838952136413576, −2.13613369195081718958016151745, −0.58205029295005054361915675698,
0.70527648633640025926086707533, 1.94794280284226063193648054999, 2.84408796788337943569852820654, 4.24798696746050213675180341069, 5.11811157095510307447535741583, 6.02027704637512071074884145592, 7.47179668472108951241009941877, 8.277632745997369497731171731419, 9.034066771684810456282343748646, 9.654057546904896436154307295182
