L(s) = 1 | + (−0.5 − 0.866i)2-s + (2.81 + 1.62i)3-s + (−0.499 + 0.866i)4-s + (−1.55 − 2.68i)5-s − 3.24i·6-s + (0.334 + 2.62i)7-s + 0.999·8-s + (3.77 + 6.54i)9-s + (−1.55 + 2.68i)10-s + (4.07 + 2.35i)11-s + (−2.81 + 1.62i)12-s − 3.22i·13-s + (2.10 − 1.60i)14-s − 10.0i·15-s + (−0.5 − 0.866i)16-s + (−0.465 + 0.806i)17-s + ⋯ |
L(s) = 1 | + (−0.353 − 0.612i)2-s + (1.62 + 0.937i)3-s + (−0.249 + 0.433i)4-s + (−0.693 − 1.20i)5-s − 1.32i·6-s + (0.126 + 0.991i)7-s + 0.353·8-s + (1.25 + 2.18i)9-s + (−0.490 + 0.849i)10-s + (1.22 + 0.709i)11-s + (−0.812 + 0.468i)12-s − 0.894i·13-s + (0.562 − 0.428i)14-s − 2.60i·15-s + (−0.125 − 0.216i)16-s + (−0.112 + 0.195i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 322 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.998 - 0.0588i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 322 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.998 - 0.0588i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.70382 + 0.0502013i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.70382 + 0.0502013i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.5 + 0.866i)T \) |
| 7 | \( 1 + (-0.334 - 2.62i)T \) |
| 23 | \( 1 + (-4.53 + 1.55i)T \) |
good | 3 | \( 1 + (-2.81 - 1.62i)T + (1.5 + 2.59i)T^{2} \) |
| 5 | \( 1 + (1.55 + 2.68i)T + (-2.5 + 4.33i)T^{2} \) |
| 11 | \( 1 + (-4.07 - 2.35i)T + (5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + 3.22iT - 13T^{2} \) |
| 17 | \( 1 + (0.465 - 0.806i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (1.55 + 2.68i)T + (-9.5 + 16.4i)T^{2} \) |
| 29 | \( 1 - 1.03T + 29T^{2} \) |
| 31 | \( 1 + (8.51 + 4.91i)T + (15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (9.60 - 5.54i)T + (18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + 3.22iT - 41T^{2} \) |
| 43 | \( 1 + 0.943iT - 43T^{2} \) |
| 47 | \( 1 + (-4.29 + 2.47i)T + (23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (6.11 + 3.52i)T + (26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (-0.109 - 0.0634i)T + (29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (-2.63 - 4.56i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (-0.0606 - 0.0349i)T + (33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + 5.66T + 71T^{2} \) |
| 73 | \( 1 + (8.51 + 4.91i)T + (36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (-1.39 + 0.806i)T + (39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 - 1.34T + 83T^{2} \) |
| 89 | \( 1 + (6.59 + 11.4i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 - 7.02T + 97T^{2} \) |
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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
−11.65462786891569221055907651203, −10.46187566266557026092607069916, −9.402083566351775841831538107856, −8.840558898844439087436788520136, −8.487373707282008208069650157122, −7.38399067468069358113386186081, −5.06412954995623155958105905108, −4.24934749833653341808816699875, −3.24321409162123317927561920665, −1.88883282421863559594962965813,
1.53628269164787449815281827091, 3.31038131084461629113161319396, 3.97733253995919299188572045665, 6.51375533045192121612890900453, 7.05674070852002211185852264039, 7.58488176679917660142541657833, 8.672213268519646167909347782287, 9.298222886942018060831430523056, 10.60541887928138278343701756079, 11.54445451082962254915922944192