| L(s) = 1 | + (4.50 + 7.79i)3-s + (20.8 − 12.0i)5-s + (37.9 − 65.6i)7-s + (−40.4 + 70.1i)9-s + (64.0 + 36.9i)11-s + (119. + 207. i)13-s + (187. + 108. i)15-s − 120. i·17-s − 237.·19-s + (682. − 0.297i)21-s + (279. − 161. i)23-s + (−23.0 + 39.8i)25-s + (−728. + 0.952i)27-s + (1.24e3 + 718. i)29-s + (−762. − 1.32e3i)31-s + ⋯ |
| L(s) = 1 | + (0.500 + 0.865i)3-s + (0.833 − 0.481i)5-s + (0.773 − 1.34i)7-s + (−0.499 + 0.866i)9-s + (0.529 + 0.305i)11-s + (0.709 + 1.22i)13-s + (0.833 + 0.480i)15-s − 0.416i·17-s − 0.657·19-s + (1.54 − 0.000673i)21-s + (0.529 − 0.305i)23-s + (−0.0368 + 0.0638i)25-s + (−0.999 + 0.00130i)27-s + (1.48 + 0.854i)29-s + (−0.793 − 1.37i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 72 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.939 - 0.342i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 72 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.939 - 0.342i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(2.28088 + 0.403205i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.28088 + 0.403205i\) |
| \(L(3)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (-4.50 - 7.79i)T \) |
| good | 5 | \( 1 + (-20.8 + 12.0i)T + (312.5 - 541. i)T^{2} \) |
| 7 | \( 1 + (-37.9 + 65.6i)T + (-1.20e3 - 2.07e3i)T^{2} \) |
| 11 | \( 1 + (-64.0 - 36.9i)T + (7.32e3 + 1.26e4i)T^{2} \) |
| 13 | \( 1 + (-119. - 207. i)T + (-1.42e4 + 2.47e4i)T^{2} \) |
| 17 | \( 1 + 120. iT - 8.35e4T^{2} \) |
| 19 | \( 1 + 237.T + 1.30e5T^{2} \) |
| 23 | \( 1 + (-279. + 161. i)T + (1.39e5 - 2.42e5i)T^{2} \) |
| 29 | \( 1 + (-1.24e3 - 718. i)T + (3.53e5 + 6.12e5i)T^{2} \) |
| 31 | \( 1 + (762. + 1.32e3i)T + (-4.61e5 + 7.99e5i)T^{2} \) |
| 37 | \( 1 + 2.18e3T + 1.87e6T^{2} \) |
| 41 | \( 1 + (-311. + 179. i)T + (1.41e6 - 2.44e6i)T^{2} \) |
| 43 | \( 1 + (633. - 1.09e3i)T + (-1.70e6 - 2.96e6i)T^{2} \) |
| 47 | \( 1 + (2.91e3 + 1.68e3i)T + (2.43e6 + 4.22e6i)T^{2} \) |
| 53 | \( 1 + 1.88e3iT - 7.89e6T^{2} \) |
| 59 | \( 1 + (-4.18e3 + 2.41e3i)T + (6.05e6 - 1.04e7i)T^{2} \) |
| 61 | \( 1 + (1.82e3 - 3.16e3i)T + (-6.92e6 - 1.19e7i)T^{2} \) |
| 67 | \( 1 + (1.78e3 + 3.08e3i)T + (-1.00e7 + 1.74e7i)T^{2} \) |
| 71 | \( 1 - 3.49e3iT - 2.54e7T^{2} \) |
| 73 | \( 1 + 2.66e3T + 2.83e7T^{2} \) |
| 79 | \( 1 + (407. - 705. i)T + (-1.94e7 - 3.37e7i)T^{2} \) |
| 83 | \( 1 + (-306. - 177. i)T + (2.37e7 + 4.11e7i)T^{2} \) |
| 89 | \( 1 - 8.10e3iT - 6.27e7T^{2} \) |
| 97 | \( 1 + (2.46e3 - 4.26e3i)T + (-4.42e7 - 7.66e7i)T^{2} \) |
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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
−14.01894262864617650191599225276, −13.29184181312620291413328232426, −11.45227251437612758765356003079, −10.47271101169200210778151428933, −9.412518790334316898783132472377, −8.442488731770191933738045518256, −6.83923939943481433763940901917, −4.96349170030808564770777303955, −3.95822244515432729638282012687, −1.65859162370950311617160160519,
1.64740515528054856051448055194, 2.98338992934716039032338176037, 5.58146595787746523587134057391, 6.51634131905521194208771231718, 8.226139961367521248474478418469, 8.899678182849294776395718810655, 10.50553134258853821174287322934, 11.82067300169436084095483700056, 12.79512577263603623524056468579, 13.89859200479793719925831351454
