| L(s) = 1 | − 46.7i·3-s + 704.·5-s − 2.00e3i·7-s − 2.18e3·9-s − 4.08e3i·11-s − 2.78e4·13-s − 3.29e4i·15-s + 4.58e3·17-s + 1.32e5i·19-s − 9.39e4·21-s + 4.97e5i·23-s + 1.05e5·25-s + 1.02e5i·27-s + 8.99e4·29-s + 7.32e4i·31-s + ⋯ |
| L(s) = 1 | − 0.577i·3-s + 1.12·5-s − 0.836i·7-s − 0.333·9-s − 0.278i·11-s − 0.974·13-s − 0.650i·15-s + 0.0548·17-s + 1.01i·19-s − 0.482·21-s + 1.77i·23-s + 0.270·25-s + 0.192i·27-s + 0.127·29-s + 0.0793i·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
| \(L(\frac{9}{2})\) |
\(\approx\) |
\(2.510994061\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.510994061\) |
| \(L(5)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + 46.7iT \) |
| good | 5 | \( 1 - 704.T + 3.90e5T^{2} \) |
| 7 | \( 1 + 2.00e3iT - 5.76e6T^{2} \) |
| 11 | \( 1 + 4.08e3iT - 2.14e8T^{2} \) |
| 13 | \( 1 + 2.78e4T + 8.15e8T^{2} \) |
| 17 | \( 1 - 4.58e3T + 6.97e9T^{2} \) |
| 19 | \( 1 - 1.32e5iT - 1.69e10T^{2} \) |
| 23 | \( 1 - 4.97e5iT - 7.83e10T^{2} \) |
| 29 | \( 1 - 8.99e4T + 5.00e11T^{2} \) |
| 31 | \( 1 - 7.32e4iT - 8.52e11T^{2} \) |
| 37 | \( 1 - 1.83e6T + 3.51e12T^{2} \) |
| 41 | \( 1 - 4.67e6T + 7.98e12T^{2} \) |
| 43 | \( 1 + 4.30e5iT - 1.16e13T^{2} \) |
| 47 | \( 1 - 8.22e6iT - 2.38e13T^{2} \) |
| 53 | \( 1 - 3.66e6T + 6.22e13T^{2} \) |
| 59 | \( 1 - 7.42e6iT - 1.46e14T^{2} \) |
| 61 | \( 1 - 1.55e7T + 1.91e14T^{2} \) |
| 67 | \( 1 + 1.00e7iT - 4.06e14T^{2} \) |
| 71 | \( 1 + 5.70e6iT - 6.45e14T^{2} \) |
| 73 | \( 1 + 3.27e7T + 8.06e14T^{2} \) |
| 79 | \( 1 + 4.74e6iT - 1.51e15T^{2} \) |
| 83 | \( 1 - 1.31e7iT - 2.25e15T^{2} \) |
| 89 | \( 1 - 6.90e7T + 3.93e15T^{2} \) |
| 97 | \( 1 + 1.35e8T + 7.83e15T^{2} \) |
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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
−9.877773840718550376452529260300, −9.296983897352580560758850567378, −7.87802362765249007890433442934, −7.28433835548642610669865612013, −6.11854497086659437713042325978, −5.46682057282178254164323456486, −4.11380431871896417421904599324, −2.80632799049779533076958197043, −1.74194763066897073355278384024, −0.877121306771261804373314660322,
0.55440611058575124068985409796, 2.24306820991991044995717785739, 2.63775531967755558642309042228, 4.34386036015392204529371138487, 5.21262999737737013538629027289, 6.03000768063610794304313605804, 7.03382832054559211321999639184, 8.419138999906415263263433532197, 9.252624888041204927923709580245, 9.862291962480257585079672102727
