L(s) = 1 | + (39.2 − 67.9i)5-s + (110. + 191. i)7-s + (−115. − 199. i)11-s + (385. − 668. i)13-s + 769.·17-s + 383.·19-s + (−193. + 334. i)23-s + (−1.51e3 − 2.62e3i)25-s + (−394. − 683. i)29-s + (1.60e3 − 2.78e3i)31-s + 1.73e4·35-s + 2.46e3·37-s + (−4.62e3 + 8.00e3i)41-s + (5.31e3 + 9.20e3i)43-s + (−976. − 1.69e3i)47-s + ⋯ |
L(s) = 1 | + (0.701 − 1.21i)5-s + (0.852 + 1.47i)7-s + (−0.286 − 0.496i)11-s + (0.633 − 1.09i)13-s + 0.646·17-s + 0.243·19-s + (−0.0761 + 0.131i)23-s + (−0.485 − 0.841i)25-s + (−0.0871 − 0.151i)29-s + (0.300 − 0.521i)31-s + 2.39·35-s + 0.296·37-s + (−0.429 + 0.743i)41-s + (0.438 + 0.759i)43-s + (−0.0644 − 0.111i)47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 432 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.687 + 0.726i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 432 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.687 + 0.726i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(3)\) |
\(\approx\) |
\(2.904193892\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.904193892\) |
\(L(\frac{7}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
good | 5 | \( 1 + (-39.2 + 67.9i)T + (-1.56e3 - 2.70e3i)T^{2} \) |
| 7 | \( 1 + (-110. - 191. i)T + (-8.40e3 + 1.45e4i)T^{2} \) |
| 11 | \( 1 + (115. + 199. i)T + (-8.05e4 + 1.39e5i)T^{2} \) |
| 13 | \( 1 + (-385. + 668. i)T + (-1.85e5 - 3.21e5i)T^{2} \) |
| 17 | \( 1 - 769.T + 1.41e6T^{2} \) |
| 19 | \( 1 - 383.T + 2.47e6T^{2} \) |
| 23 | \( 1 + (193. - 334. i)T + (-3.21e6 - 5.57e6i)T^{2} \) |
| 29 | \( 1 + (394. + 683. i)T + (-1.02e7 + 1.77e7i)T^{2} \) |
| 31 | \( 1 + (-1.60e3 + 2.78e3i)T + (-1.43e7 - 2.47e7i)T^{2} \) |
| 37 | \( 1 - 2.46e3T + 6.93e7T^{2} \) |
| 41 | \( 1 + (4.62e3 - 8.00e3i)T + (-5.79e7 - 1.00e8i)T^{2} \) |
| 43 | \( 1 + (-5.31e3 - 9.20e3i)T + (-7.35e7 + 1.27e8i)T^{2} \) |
| 47 | \( 1 + (976. + 1.69e3i)T + (-1.14e8 + 1.98e8i)T^{2} \) |
| 53 | \( 1 - 3.25e4T + 4.18e8T^{2} \) |
| 59 | \( 1 + (-1.19e4 + 2.06e4i)T + (-3.57e8 - 6.19e8i)T^{2} \) |
| 61 | \( 1 + (1.88e4 + 3.25e4i)T + (-4.22e8 + 7.31e8i)T^{2} \) |
| 67 | \( 1 + (1.15e4 - 1.99e4i)T + (-6.75e8 - 1.16e9i)T^{2} \) |
| 71 | \( 1 + 6.60e4T + 1.80e9T^{2} \) |
| 73 | \( 1 - 6.51e4T + 2.07e9T^{2} \) |
| 79 | \( 1 + (3.54e4 + 6.13e4i)T + (-1.53e9 + 2.66e9i)T^{2} \) |
| 83 | \( 1 + (2.76e4 + 4.78e4i)T + (-1.96e9 + 3.41e9i)T^{2} \) |
| 89 | \( 1 + 1.05e4T + 5.58e9T^{2} \) |
| 97 | \( 1 + (-4.14e4 - 7.17e4i)T + (-4.29e9 + 7.43e9i)T^{2} \) |
show more | |
show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.08746057400290611007436588791, −9.162772498078115576658138290037, −8.446295960437982884442062473803, −7.88626450838342323794420193226, −5.92806946163499283730790830126, −5.57815174729452483168412642674, −4.72507438870421166561144032576, −3.04278717962441169952691306322, −1.83173223063805881320396583616, −0.799715271963220810157954565059,
1.12740245461596870478985412935, 2.19389548430297005113077299167, 3.57768335753677543583466105826, 4.54223309721499718662224313658, 5.84260214372578682742685448798, 7.01063541678174098225572093235, 7.33387412514136932020890166619, 8.616088410771564432903415649781, 9.904866491201269732822897522908, 10.46439440860496512686744582457