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
−11.54445451082962254915922944192, −10.60541887928138278343701756079, −9.298222886942018060831430523056, −8.672213268519646167909347782287, −7.58488176679917660142541657833, −7.05674070852002211185852264039, −6.51375533045192121612890900453, −3.97733253995919299188572045665, −3.31038131084461629113161319396, −1.53628269164787449815281827091,
1.88883282421863559594962965813, 3.24321409162123317927561920665, 4.24934749833653341808816699875, 5.06412954995623155958105905108, 7.38399067468069358113386186081, 8.487373707282008208069650157122, 8.840558898844439087436788520136, 9.402083566351775841831538107856, 10.46187566266557026092607069916, 11.65462786891569221055907651203