| L(s) = 1 | + (0.841 − 0.540i)2-s + (−0.142 + 0.989i)3-s + (0.415 − 0.909i)4-s + (−3.15 + 0.927i)5-s + (0.415 + 0.909i)6-s + (0.654 + 0.755i)7-s + (−0.142 − 0.989i)8-s + (−0.959 − 0.281i)9-s + (−2.15 + 2.48i)10-s + (−3.69 − 2.37i)11-s + (0.841 + 0.540i)12-s + (4.43 − 5.12i)13-s + (0.959 + 0.281i)14-s + (−0.468 − 3.25i)15-s + (−0.654 − 0.755i)16-s + (0.187 + 0.410i)17-s + ⋯ |
| L(s) = 1 | + (0.594 − 0.382i)2-s + (−0.0821 + 0.571i)3-s + (0.207 − 0.454i)4-s + (−1.41 + 0.414i)5-s + (0.169 + 0.371i)6-s + (0.247 + 0.285i)7-s + (−0.0503 − 0.349i)8-s + (−0.319 − 0.0939i)9-s + (−0.681 + 0.786i)10-s + (−1.11 − 0.716i)11-s + (0.242 + 0.156i)12-s + (1.23 − 1.42i)13-s + (0.256 + 0.0752i)14-s + (−0.120 − 0.841i)15-s + (−0.163 − 0.188i)16-s + (0.0454 + 0.0994i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 966 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.354 + 0.934i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 966 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.354 + 0.934i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.580783 - 0.841737i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.580783 - 0.841737i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.841 + 0.540i)T \) |
| 3 | \( 1 + (0.142 - 0.989i)T \) |
| 7 | \( 1 + (-0.654 - 0.755i)T \) |
| 23 | \( 1 + (3.73 - 3.00i)T \) |
| good | 5 | \( 1 + (3.15 - 0.927i)T + (4.20 - 2.70i)T^{2} \) |
| 11 | \( 1 + (3.69 + 2.37i)T + (4.56 + 10.0i)T^{2} \) |
| 13 | \( 1 + (-4.43 + 5.12i)T + (-1.85 - 12.8i)T^{2} \) |
| 17 | \( 1 + (-0.187 - 0.410i)T + (-11.1 + 12.8i)T^{2} \) |
| 19 | \( 1 + (-0.658 + 1.44i)T + (-12.4 - 14.3i)T^{2} \) |
| 29 | \( 1 + (1.91 + 4.18i)T + (-18.9 + 21.9i)T^{2} \) |
| 31 | \( 1 + (1.15 + 8.00i)T + (-29.7 + 8.73i)T^{2} \) |
| 37 | \( 1 + (4.51 + 1.32i)T + (31.1 + 20.0i)T^{2} \) |
| 41 | \( 1 + (-6.60 + 1.94i)T + (34.4 - 22.1i)T^{2} \) |
| 43 | \( 1 + (-1.86 + 12.9i)T + (-41.2 - 12.1i)T^{2} \) |
| 47 | \( 1 - 3.05T + 47T^{2} \) |
| 53 | \( 1 + (3.67 + 4.24i)T + (-7.54 + 52.4i)T^{2} \) |
| 59 | \( 1 + (-1.03 + 1.19i)T + (-8.39 - 58.3i)T^{2} \) |
| 61 | \( 1 + (-0.595 - 4.14i)T + (-58.5 + 17.1i)T^{2} \) |
| 67 | \( 1 + (8.79 - 5.65i)T + (27.8 - 60.9i)T^{2} \) |
| 71 | \( 1 + (4.69 - 3.01i)T + (29.4 - 64.5i)T^{2} \) |
| 73 | \( 1 + (6.28 - 13.7i)T + (-47.8 - 55.1i)T^{2} \) |
| 79 | \( 1 + (0.992 - 1.14i)T + (-11.2 - 78.1i)T^{2} \) |
| 83 | \( 1 + (-0.566 - 0.166i)T + (69.8 + 44.8i)T^{2} \) |
| 89 | \( 1 + (-1.79 + 12.4i)T + (-85.3 - 25.0i)T^{2} \) |
| 97 | \( 1 + (9.89 - 2.90i)T + (81.6 - 52.4i)T^{2} \) |
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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
−10.18451779895186390122567388965, −8.818865447558243013207057564211, −8.035494099565650220810241740684, −7.44231276220404889380925976440, −5.85279385904709001736359673989, −5.48585264179385678084075795395, −4.10000541458569352914400210667, −3.56410144748514439905639101499, −2.63032288221616276761162524645, −0.39677549621598553728771960142,
1.59379723761980692307581303036, 3.19183267511649643033139920160, 4.25536184825219428714520483375, 4.81357514502098109157614788823, 6.07361383577910065531034608754, 7.00511320431518081503143622930, 7.69056277709896379007462429841, 8.273897547046813962286147963730, 9.116418073751440649647344733584, 10.63403780151639600715010500236
