| L(s) = 1 | + (0.939 − 1.45i)3-s + (−1.07 + 0.188i)5-s + (0.0466 − 0.0555i)7-s + (−1.23 − 2.73i)9-s + (0.889 − 5.04i)11-s + (−5.31 − 1.93i)13-s + (−0.730 + 1.73i)15-s + (3.79 − 2.19i)17-s + (4.96 + 2.86i)19-s + (−0.0370 − 0.119i)21-s + (3.14 − 2.64i)23-s + (−3.58 + 1.30i)25-s + (−5.13 − 0.767i)27-s + (−1.30 − 3.58i)29-s + (2.61 + 3.11i)31-s + ⋯ |
| L(s) = 1 | + (0.542 − 0.840i)3-s + (−0.478 + 0.0844i)5-s + (0.0176 − 0.0209i)7-s + (−0.412 − 0.911i)9-s + (0.268 − 1.52i)11-s + (−1.47 − 0.536i)13-s + (−0.188 + 0.448i)15-s + (0.920 − 0.531i)17-s + (1.13 + 0.657i)19-s + (−0.00808 − 0.0261i)21-s + (0.656 − 0.550i)23-s + (−0.717 + 0.261i)25-s + (−0.989 − 0.147i)27-s + (−0.242 − 0.665i)29-s + (0.468 + 0.558i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 432 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.205 + 0.978i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 432 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.205 + 0.978i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.863405 - 1.06300i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.863405 - 1.06300i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (-0.939 + 1.45i)T \) |
| good | 5 | \( 1 + (1.07 - 0.188i)T + (4.69 - 1.71i)T^{2} \) |
| 7 | \( 1 + (-0.0466 + 0.0555i)T + (-1.21 - 6.89i)T^{2} \) |
| 11 | \( 1 + (-0.889 + 5.04i)T + (-10.3 - 3.76i)T^{2} \) |
| 13 | \( 1 + (5.31 + 1.93i)T + (9.95 + 8.35i)T^{2} \) |
| 17 | \( 1 + (-3.79 + 2.19i)T + (8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (-4.96 - 2.86i)T + (9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (-3.14 + 2.64i)T + (3.99 - 22.6i)T^{2} \) |
| 29 | \( 1 + (1.30 + 3.58i)T + (-22.2 + 18.6i)T^{2} \) |
| 31 | \( 1 + (-2.61 - 3.11i)T + (-5.38 + 30.5i)T^{2} \) |
| 37 | \( 1 + (1.14 + 1.98i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (0.494 - 1.35i)T + (-31.4 - 26.3i)T^{2} \) |
| 43 | \( 1 + (0.128 + 0.0227i)T + (40.4 + 14.7i)T^{2} \) |
| 47 | \( 1 + (-4.26 - 3.57i)T + (8.16 + 46.2i)T^{2} \) |
| 53 | \( 1 - 10.4iT - 53T^{2} \) |
| 59 | \( 1 + (1.69 + 9.59i)T + (-55.4 + 20.1i)T^{2} \) |
| 61 | \( 1 + (-4.96 - 4.16i)T + (10.5 + 60.0i)T^{2} \) |
| 67 | \( 1 + (-2.28 + 6.27i)T + (-51.3 - 43.0i)T^{2} \) |
| 71 | \( 1 + (3.47 + 6.02i)T + (-35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + (-2.77 + 4.80i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (-4.83 - 13.2i)T + (-60.5 + 50.7i)T^{2} \) |
| 83 | \( 1 + (-3.77 + 1.37i)T + (63.5 - 53.3i)T^{2} \) |
| 89 | \( 1 + (-14.4 - 8.34i)T + (44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (2.74 - 15.5i)T + (-91.1 - 33.1i)T^{2} \) |
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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
−11.08619410105186812129480553188, −9.842477928456056028936699033254, −9.001534493838347752327262699024, −7.80082077177925053049768005634, −7.59061877702820495658734283566, −6.26146611443148349509989536649, −5.26196527494168848197109840579, −3.56031787658944474676081527844, −2.75267961710248200511759343163, −0.844704991097303548589962971227,
2.16056295332672546349227273782, 3.52000964171218268320953018528, 4.58578601465351925689244605029, 5.30059758249189609675215000584, 7.14023635171111174883312190133, 7.63310300810646173192811333173, 8.861448196803716324535133056320, 9.795512121529721313510848833455, 10.08481899400527943680139727605, 11.55351710979233352936714152471
