| L(s) = 1 | − 5.48i·3-s + (53.0 + 17.7i)5-s − 188. i·7-s + 212.·9-s + 501.·11-s + 1.06e3i·13-s + (97.5 − 290. i)15-s − 29.5i·17-s + 1.57e3·19-s − 1.03e3·21-s + 1.29e3i·23-s + (2.49e3 + 1.88e3i)25-s − 2.50e3i·27-s − 3.58e3·29-s − 3.52e3·31-s + ⋯ |
| L(s) = 1 | − 0.352i·3-s + (0.948 + 0.317i)5-s − 1.45i·7-s + 0.875·9-s + 1.25·11-s + 1.74i·13-s + (0.111 − 0.333i)15-s − 0.0248i·17-s + 1.00·19-s − 0.513·21-s + 0.510i·23-s + (0.798 + 0.602i)25-s − 0.660i·27-s − 0.791·29-s − 0.659·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 320 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.948 + 0.317i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 320 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.948 + 0.317i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(3.120728137\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.120728137\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 + (-53.0 - 17.7i)T \) |
| good | 3 | \( 1 + 5.48iT - 243T^{2} \) |
| 7 | \( 1 + 188. iT - 1.68e4T^{2} \) |
| 11 | \( 1 - 501.T + 1.61e5T^{2} \) |
| 13 | \( 1 - 1.06e3iT - 3.71e5T^{2} \) |
| 17 | \( 1 + 29.5iT - 1.41e6T^{2} \) |
| 19 | \( 1 - 1.57e3T + 2.47e6T^{2} \) |
| 23 | \( 1 - 1.29e3iT - 6.43e6T^{2} \) |
| 29 | \( 1 + 3.58e3T + 2.05e7T^{2} \) |
| 31 | \( 1 + 3.52e3T + 2.86e7T^{2} \) |
| 37 | \( 1 + 8.41e3iT - 6.93e7T^{2} \) |
| 41 | \( 1 - 7.01e3T + 1.15e8T^{2} \) |
| 43 | \( 1 - 2.26e4iT - 1.47e8T^{2} \) |
| 47 | \( 1 + 3.50e3iT - 2.29e8T^{2} \) |
| 53 | \( 1 - 2.73e4iT - 4.18e8T^{2} \) |
| 59 | \( 1 - 7.92e3T + 7.14e8T^{2} \) |
| 61 | \( 1 - 7.02e3T + 8.44e8T^{2} \) |
| 67 | \( 1 + 1.76e4iT - 1.35e9T^{2} \) |
| 71 | \( 1 - 1.34e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 3.99e4iT - 2.07e9T^{2} \) |
| 79 | \( 1 - 9.33e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + 5.84e4iT - 3.93e9T^{2} \) |
| 89 | \( 1 - 1.39e4T + 5.58e9T^{2} \) |
| 97 | \( 1 + 1.10e5iT - 8.58e9T^{2} \) |
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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
−10.77691648912305896859637642383, −9.551656701768483531445567207716, −9.358372337969513176797100263937, −7.44346197171841642916705740582, −6.98990527053543672241596604787, −6.12935541128107659583575004588, −4.51634894837720179094761513677, −3.68043330871409290085833821825, −1.80008508909691968244682749483, −1.12921163313795282192718778988,
1.05419055146902240813976103219, 2.27778683286772488784550700867, 3.58383262560033827697317904648, 5.14211558255885672054951842342, 5.68879145519129765151770323853, 6.82313492890486375246897952468, 8.229979366915336195616627796983, 9.199384665345287373964807446812, 9.716184812006857286295335080611, 10.67747509734756700068493096172
