| L(s) = 1 | + (−0.415 + 0.909i)2-s + (−0.959 − 0.281i)3-s + (−0.654 − 0.755i)4-s + (−0.117 + 0.0754i)5-s + (0.654 − 0.755i)6-s + (−0.142 + 0.989i)7-s + (0.959 − 0.281i)8-s + (0.841 + 0.540i)9-s + (−0.0198 − 0.138i)10-s + (−1.36 − 2.98i)11-s + (0.415 + 0.909i)12-s + (0.379 + 2.64i)13-s + (−0.841 − 0.540i)14-s + (0.133 − 0.0392i)15-s + (−0.142 + 0.989i)16-s + (0.888 − 1.02i)17-s + ⋯ |
| L(s) = 1 | + (−0.293 + 0.643i)2-s + (−0.553 − 0.162i)3-s + (−0.327 − 0.377i)4-s + (−0.0524 + 0.0337i)5-s + (0.267 − 0.308i)6-s + (−0.0537 + 0.374i)7-s + (0.339 − 0.0996i)8-s + (0.280 + 0.180i)9-s + (−0.00627 − 0.0436i)10-s + (−0.410 − 0.899i)11-s + (0.119 + 0.262i)12-s + (0.105 + 0.732i)13-s + (−0.224 − 0.144i)14-s + (0.0345 − 0.0101i)15-s + (−0.0355 + 0.247i)16-s + (0.215 − 0.248i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 966 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.227 + 0.973i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 966 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.227 + 0.973i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.210717 - 0.265614i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.210717 - 0.265614i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.415 - 0.909i)T \) |
| 3 | \( 1 + (0.959 + 0.281i)T \) |
| 7 | \( 1 + (0.142 - 0.989i)T \) |
| 23 | \( 1 + (2.46 - 4.11i)T \) |
| good | 5 | \( 1 + (0.117 - 0.0754i)T + (2.07 - 4.54i)T^{2} \) |
| 11 | \( 1 + (1.36 + 2.98i)T + (-7.20 + 8.31i)T^{2} \) |
| 13 | \( 1 + (-0.379 - 2.64i)T + (-12.4 + 3.66i)T^{2} \) |
| 17 | \( 1 + (-0.888 + 1.02i)T + (-2.41 - 16.8i)T^{2} \) |
| 19 | \( 1 + (3.67 + 4.24i)T + (-2.70 + 18.8i)T^{2} \) |
| 29 | \( 1 + (-3.00 + 3.46i)T + (-4.12 - 28.7i)T^{2} \) |
| 31 | \( 1 + (-1.90 + 0.560i)T + (26.0 - 16.7i)T^{2} \) |
| 37 | \( 1 + (8.05 + 5.17i)T + (15.3 + 33.6i)T^{2} \) |
| 41 | \( 1 + (-5.91 + 3.79i)T + (17.0 - 37.2i)T^{2} \) |
| 43 | \( 1 + (10.7 + 3.16i)T + (36.1 + 23.2i)T^{2} \) |
| 47 | \( 1 + 2.44T + 47T^{2} \) |
| 53 | \( 1 + (-0.943 + 6.56i)T + (-50.8 - 14.9i)T^{2} \) |
| 59 | \( 1 + (2.11 + 14.7i)T + (-56.6 + 16.6i)T^{2} \) |
| 61 | \( 1 + (6.29 - 1.84i)T + (51.3 - 32.9i)T^{2} \) |
| 67 | \( 1 + (0.751 - 1.64i)T + (-43.8 - 50.6i)T^{2} \) |
| 71 | \( 1 + (-3.34 + 7.31i)T + (-46.4 - 53.6i)T^{2} \) |
| 73 | \( 1 + (4.55 + 5.25i)T + (-10.3 + 72.2i)T^{2} \) |
| 79 | \( 1 + (0.319 + 2.22i)T + (-75.7 + 22.2i)T^{2} \) |
| 83 | \( 1 + (3.86 + 2.48i)T + (34.4 + 75.4i)T^{2} \) |
| 89 | \( 1 + (9.43 + 2.76i)T + (74.8 + 48.1i)T^{2} \) |
| 97 | \( 1 + (-4.29 + 2.76i)T + (40.2 - 88.2i)T^{2} \) |
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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
−9.644055730220779091006694877573, −8.873539515500301555897788498606, −8.087931201727186252739091215710, −7.15572448974668299345842016196, −6.36922430071666988377493045377, −5.60164014771348644741319881680, −4.77870013501080698977971756567, −3.51031470266871223586997396812, −1.93532728727300447664003613917, −0.18905265505344206297284901013,
1.46470959837304295049780766847, 2.82486496431992930595142590129, 4.07427594802255730698730071878, 4.78748525506119478966375448209, 5.94849272055378021416069545549, 6.87847750616932103080712892685, 7.970091448372077741912001954854, 8.525240230201329181443239296574, 9.894315285951409166078044608840, 10.26047882428905551669528162804
