| L(s) = 1 | + (1.38 − 0.270i)2-s + (1.18 − 2.60i)3-s + (1.85 − 0.750i)4-s + (1.05 + 1.97i)5-s + (0.945 − 3.93i)6-s + (1.18 + 1.84i)7-s + (2.37 − 1.54i)8-s + (−3.39 − 3.91i)9-s + (1.99 + 2.45i)10-s + (−0.231 − 0.0333i)11-s + (0.249 − 5.71i)12-s + (−1.88 − 1.21i)13-s + (2.14 + 2.24i)14-s + (6.38 − 0.394i)15-s + (2.87 − 2.78i)16-s + (−1.92 + 6.54i)17-s + ⋯ |
| L(s) = 1 | + (0.981 − 0.191i)2-s + (0.685 − 1.50i)3-s + (0.926 − 0.375i)4-s + (0.470 + 0.882i)5-s + (0.386 − 1.60i)6-s + (0.449 + 0.698i)7-s + (0.837 − 0.545i)8-s + (−1.13 − 1.30i)9-s + (0.630 + 0.775i)10-s + (−0.0699 − 0.0100i)11-s + (0.0719 − 1.64i)12-s + (−0.523 − 0.336i)13-s + (0.574 + 0.600i)14-s + (1.64 − 0.101i)15-s + (0.718 − 0.695i)16-s + (−0.465 + 1.58i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 920 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.366 + 0.930i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 920 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.366 + 0.930i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.27080 - 2.22701i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.27080 - 2.22701i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-1.38 + 0.270i)T \) |
| 5 | \( 1 + (-1.05 - 1.97i)T \) |
| 23 | \( 1 + (-4.77 - 0.402i)T \) |
| good | 3 | \( 1 + (-1.18 + 2.60i)T + (-1.96 - 2.26i)T^{2} \) |
| 7 | \( 1 + (-1.18 - 1.84i)T + (-2.90 + 6.36i)T^{2} \) |
| 11 | \( 1 + (0.231 + 0.0333i)T + (10.5 + 3.09i)T^{2} \) |
| 13 | \( 1 + (1.88 + 1.21i)T + (5.40 + 11.8i)T^{2} \) |
| 17 | \( 1 + (1.92 - 6.54i)T + (-14.3 - 9.19i)T^{2} \) |
| 19 | \( 1 + (1.92 + 6.53i)T + (-15.9 + 10.2i)T^{2} \) |
| 29 | \( 1 + (-1.03 + 3.51i)T + (-24.3 - 15.6i)T^{2} \) |
| 31 | \( 1 + (3.95 + 8.66i)T + (-20.3 + 23.4i)T^{2} \) |
| 37 | \( 1 + (-3.44 - 3.98i)T + (-5.26 + 36.6i)T^{2} \) |
| 41 | \( 1 + (6.86 - 7.92i)T + (-5.83 - 40.5i)T^{2} \) |
| 43 | \( 1 + (1.97 - 4.31i)T + (-28.1 - 32.4i)T^{2} \) |
| 47 | \( 1 - 7.47iT - 47T^{2} \) |
| 53 | \( 1 + (3.29 - 2.12i)T + (22.0 - 48.2i)T^{2} \) |
| 59 | \( 1 + (2.74 - 4.26i)T + (-24.5 - 53.6i)T^{2} \) |
| 61 | \( 1 + (9.89 - 4.51i)T + (39.9 - 46.1i)T^{2} \) |
| 67 | \( 1 + (-1.28 - 8.96i)T + (-64.2 + 18.8i)T^{2} \) |
| 71 | \( 1 + (-1.38 - 9.60i)T + (-68.1 + 20.0i)T^{2} \) |
| 73 | \( 1 + (1.89 + 6.46i)T + (-61.4 + 39.4i)T^{2} \) |
| 79 | \( 1 + (-3.51 - 2.25i)T + (32.8 + 71.8i)T^{2} \) |
| 83 | \( 1 + (7.86 + 9.07i)T + (-11.8 + 82.1i)T^{2} \) |
| 89 | \( 1 + (-3.74 + 8.19i)T + (-58.2 - 67.2i)T^{2} \) |
| 97 | \( 1 + (0.414 + 0.359i)T + (13.8 + 96.0i)T^{2} \) |
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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
−10.06039932805258666469620488632, −8.922717740936013298242471435021, −7.940339473380969940875165816014, −7.23189990291718916778740966982, −6.39258765417808696975424141234, −5.90188336809418574598809098246, −4.57395093905219581714878434893, −3.01843338113160126326499496600, −2.46764646450809762130787569956, −1.60575172927989707837762790710,
1.93965537135580925083318753955, 3.24708488616792415274933635955, 4.10424563873379892715654655532, 5.03029610656576748212798322956, 5.16999666703213843262758808445, 6.76028623625446534987315192606, 7.75198057035796391352404867905, 8.728148944210838153773077286839, 9.358130309367430545937141653520, 10.38059946458339742675720291979
