| L(s) = 1 | + (−0.631 − 1.61i)3-s + (0.156 − 0.889i)5-s + (1.02 − 0.860i)7-s + (−2.20 + 2.03i)9-s + (−1.03 − 5.84i)11-s + (−0.904 + 0.329i)13-s + (−1.53 + 0.308i)15-s + (0.115 − 0.200i)17-s + (−0.756 − 1.31i)19-s + (−2.03 − 1.11i)21-s + (−3.42 − 2.87i)23-s + (3.93 + 1.43i)25-s + (4.67 + 2.26i)27-s + (5.21 + 1.89i)29-s + (7.24 + 6.07i)31-s + ⋯ |
| L(s) = 1 | + (−0.364 − 0.931i)3-s + (0.0701 − 0.397i)5-s + (0.387 − 0.325i)7-s + (−0.734 + 0.678i)9-s + (−0.310 − 1.76i)11-s + (−0.250 + 0.0913i)13-s + (−0.395 + 0.0795i)15-s + (0.0280 − 0.0486i)17-s + (−0.173 − 0.300i)19-s + (−0.444 − 0.242i)21-s + (−0.714 − 0.599i)23-s + (0.786 + 0.286i)25-s + (0.899 + 0.436i)27-s + (0.967 + 0.352i)29-s + (1.30 + 1.09i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 216 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.217 + 0.976i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 216 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.217 + 0.976i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.640192 - 0.798424i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.640192 - 0.798424i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (0.631 + 1.61i)T \) |
| good | 5 | \( 1 + (-0.156 + 0.889i)T + (-4.69 - 1.71i)T^{2} \) |
| 7 | \( 1 + (-1.02 + 0.860i)T + (1.21 - 6.89i)T^{2} \) |
| 11 | \( 1 + (1.03 + 5.84i)T + (-10.3 + 3.76i)T^{2} \) |
| 13 | \( 1 + (0.904 - 0.329i)T + (9.95 - 8.35i)T^{2} \) |
| 17 | \( 1 + (-0.115 + 0.200i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (0.756 + 1.31i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (3.42 + 2.87i)T + (3.99 + 22.6i)T^{2} \) |
| 29 | \( 1 + (-5.21 - 1.89i)T + (22.2 + 18.6i)T^{2} \) |
| 31 | \( 1 + (-7.24 - 6.07i)T + (5.38 + 30.5i)T^{2} \) |
| 37 | \( 1 + (-1.74 + 3.02i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (5.06 - 1.84i)T + (31.4 - 26.3i)T^{2} \) |
| 43 | \( 1 + (-1.47 - 8.38i)T + (-40.4 + 14.7i)T^{2} \) |
| 47 | \( 1 + (-5.18 + 4.35i)T + (8.16 - 46.2i)T^{2} \) |
| 53 | \( 1 - 5.69T + 53T^{2} \) |
| 59 | \( 1 + (-0.506 + 2.87i)T + (-55.4 - 20.1i)T^{2} \) |
| 61 | \( 1 + (-10.2 + 8.61i)T + (10.5 - 60.0i)T^{2} \) |
| 67 | \( 1 + (11.6 - 4.23i)T + (51.3 - 43.0i)T^{2} \) |
| 71 | \( 1 + (7.56 - 13.1i)T + (-35.5 - 61.4i)T^{2} \) |
| 73 | \( 1 + (2.83 + 4.90i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (-6.84 - 2.49i)T + (60.5 + 50.7i)T^{2} \) |
| 83 | \( 1 + (5.91 + 2.15i)T + (63.5 + 53.3i)T^{2} \) |
| 89 | \( 1 + (-7.28 - 12.6i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (2.69 + 15.2i)T + (-91.1 + 33.1i)T^{2} \) |
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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
−12.06103976873193492650574916026, −11.19390405680200889544991271448, −10.38175572873517963945373861176, −8.672104748733841264725087247558, −8.178096495814179200644325584600, −6.88226761467549455068209031417, −5.89092406405550537260473419319, −4.77632142279650691840476261536, −2.86390908501375572705387396689, −0.961102848968339201770734970429,
2.47287868676903913212799970760, 4.18571752321924125519233282145, 5.08781612970332839855998558863, 6.32556178328133914092980438580, 7.57751436139836440151430706397, 8.810451440007366371592916106417, 10.07850422439225760463891345757, 10.28663603282672537443608348071, 11.75092227499372318592968937484, 12.20616553433182661518785503427
