L(s) = 1 | + (0.866 − 0.5i)2-s + (−1.12 − 1.31i)3-s + (0.499 − 0.866i)4-s + (−1.63 − 2.83i)5-s + (−1.63 − 0.572i)6-s + (1.76 − 1.96i)7-s − 0.999i·8-s + (−0.448 + 2.96i)9-s + (−2.83 − 1.63i)10-s + (0.671 + 0.387i)11-s + (−1.70 + 0.321i)12-s − i·13-s + (0.545 − 2.58i)14-s + (−1.87 + 5.36i)15-s + (−0.5 − 0.866i)16-s + (−0.0317 + 0.0550i)17-s + ⋯ |
L(s) = 1 | + (0.612 − 0.353i)2-s + (−0.652 − 0.758i)3-s + (0.249 − 0.433i)4-s + (−0.733 − 1.27i)5-s + (−0.667 − 0.233i)6-s + (0.667 − 0.744i)7-s − 0.353i·8-s + (−0.149 + 0.988i)9-s + (−0.898 − 0.518i)10-s + (0.202 + 0.116i)11-s + (−0.491 + 0.0928i)12-s − 0.277i·13-s + (0.145 − 0.691i)14-s + (−0.484 + 1.38i)15-s + (−0.125 − 0.216i)16-s + (−0.00770 + 0.0133i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.994 + 0.105i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.994 + 0.105i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.0694363 - 1.31235i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.0694363 - 1.31235i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.866 + 0.5i)T \) |
| 3 | \( 1 + (1.12 + 1.31i)T \) |
| 7 | \( 1 + (-1.76 + 1.96i)T \) |
| 13 | \( 1 + iT \) |
good | 5 | \( 1 + (1.63 + 2.83i)T + (-2.5 + 4.33i)T^{2} \) |
| 11 | \( 1 + (-0.671 - 0.387i)T + (5.5 + 9.52i)T^{2} \) |
| 17 | \( 1 + (0.0317 - 0.0550i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (0.489 - 0.282i)T + (9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (6.86 - 3.96i)T + (11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + 4.95iT - 29T^{2} \) |
| 31 | \( 1 + (-0.453 - 0.262i)T + (15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (-4.48 - 7.77i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + 0.420T + 41T^{2} \) |
| 43 | \( 1 - 8.44T + 43T^{2} \) |
| 47 | \( 1 + (6.63 + 11.4i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (0.134 + 0.0777i)T + (26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (-3.93 + 6.81i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (1.98 - 1.14i)T + (30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (-5.40 + 9.36i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 - 0.904iT - 71T^{2} \) |
| 73 | \( 1 + (-12.4 - 7.17i)T + (36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (5.03 + 8.72i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 - 10.1T + 83T^{2} \) |
| 89 | \( 1 + (-2.77 - 4.80i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + 6.01iT - 97T^{2} \) |
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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
−10.69308206360119552808283091527, −9.677561680601144993844207449281, −8.169497322370257715226237167571, −7.84848895974353780804303851929, −6.62687299108317061476519719160, −5.49809888365040429855667257664, −4.69103011941235120832476602125, −3.87902642610913793233341198999, −1.86242014475322337110976769419, −0.69044878908389132959420473975,
2.58381840445199982519019565181, 3.77633946844752936166180603319, 4.56721302831398921094219350072, 5.76371427414914607612255185594, 6.45385609647676281885800175967, 7.46506456623886077625150269390, 8.450483836286948782380739767409, 9.558275092327966568943160038431, 10.81152013842280135583272354422, 11.11725767176374157033073173176