L(s) = 1 | + (−0.5 + 0.866i)2-s + (−0.5 − 0.866i)3-s + (−0.499 − 0.866i)4-s + (1.91 − 3.31i)5-s + 0.999·6-s + (−1.87 − 1.86i)7-s + 0.999·8-s + (−0.499 + 0.866i)9-s + (1.91 + 3.31i)10-s + (−2.63 − 4.57i)11-s + (−0.499 + 0.866i)12-s + 4.35·13-s + (2.55 − 0.690i)14-s − 3.82·15-s + (−0.5 + 0.866i)16-s + (1.23 + 2.13i)17-s + ⋯ |
L(s) = 1 | + (−0.353 + 0.612i)2-s + (−0.288 − 0.499i)3-s + (−0.249 − 0.433i)4-s + (0.856 − 1.48i)5-s + 0.408·6-s + (−0.708 − 0.705i)7-s + 0.353·8-s + (−0.166 + 0.288i)9-s + (0.605 + 1.04i)10-s + (−0.795 − 1.37i)11-s + (−0.144 + 0.249i)12-s + 1.20·13-s + (0.682 − 0.184i)14-s − 0.988·15-s + (−0.125 + 0.216i)16-s + (0.299 + 0.518i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 966 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.659 + 0.751i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 966 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.659 + 0.751i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.412010 - 0.909233i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.412010 - 0.909233i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.5 - 0.866i)T \) |
| 3 | \( 1 + (0.5 + 0.866i)T \) |
| 7 | \( 1 + (1.87 + 1.86i)T \) |
| 23 | \( 1 + (0.5 - 0.866i)T \) |
good | 5 | \( 1 + (-1.91 + 3.31i)T + (-2.5 - 4.33i)T^{2} \) |
| 11 | \( 1 + (2.63 + 4.57i)T + (-5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 - 4.35T + 13T^{2} \) |
| 17 | \( 1 + (-1.23 - 2.13i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (-0.960 + 1.66i)T + (-9.5 - 16.4i)T^{2} \) |
| 29 | \( 1 + 1.35T + 29T^{2} \) |
| 31 | \( 1 + (-0.718 - 1.24i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (3.31 - 5.74i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 - 1.92T + 41T^{2} \) |
| 43 | \( 1 + 8.21T + 43T^{2} \) |
| 47 | \( 1 + (-3.49 + 6.05i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (-1.44 - 2.50i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (2.14 + 3.72i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (5.71 - 9.90i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (2.66 + 4.61i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 - 7.64T + 71T^{2} \) |
| 73 | \( 1 + (-0.218 - 0.378i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (0.679 - 1.17i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + 4.37T + 83T^{2} \) |
| 89 | \( 1 + (-6.14 + 10.6i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 - 0.298T + 97T^{2} \) |
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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
−9.601918407336123129231392082746, −8.518028311056392940869010903264, −8.400627876768756518096665988409, −7.14084390896114129046147429812, −6.01196739075732748747052914376, −5.78070269303079462716554091966, −4.72301341662018203096732242319, −3.35821958215022254496590548441, −1.48679863622441519156311285902, −0.55657478244018201094036547041,
2.00812086188858870684426302183, 2.86527656539661304553450314180, 3.73033545897871746575920757806, 5.19085337365639316869999150861, 6.06825794259664944381746615137, 6.84695279272988831685643352987, 7.81059501578399106629148711491, 9.117632461279518259090233242432, 9.706991692007031382521702133420, 10.31598852397630248388570072950