L(s) = 1 | + (0.142 − 0.989i)2-s + (0.744 + 1.62i)3-s + (−0.959 − 0.281i)4-s + (1.17 + 1.35i)5-s + (1.71 − 0.504i)6-s + (0.841 + 0.540i)7-s + (−0.415 + 0.909i)8-s + (−0.137 + 0.158i)9-s + (1.50 − 0.969i)10-s + (0.146 + 1.02i)11-s + (−0.254 − 1.77i)12-s + (−5.05 + 3.24i)13-s + (0.654 − 0.755i)14-s + (−1.33 + 2.92i)15-s + (0.841 + 0.540i)16-s + (3.82 − 1.12i)17-s + ⋯ |
L(s) = 1 | + (0.100 − 0.699i)2-s + (0.429 + 0.940i)3-s + (−0.479 − 0.140i)4-s + (0.525 + 0.606i)5-s + (0.701 − 0.206i)6-s + (0.317 + 0.204i)7-s + (−0.146 + 0.321i)8-s + (−0.0458 + 0.0529i)9-s + (0.477 − 0.306i)10-s + (0.0442 + 0.307i)11-s + (−0.0736 − 0.511i)12-s + (−1.40 + 0.901i)13-s + (0.175 − 0.201i)14-s + (−0.344 + 0.754i)15-s + (0.210 + 0.135i)16-s + (0.927 − 0.272i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 322 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.891 - 0.453i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 322 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.891 - 0.453i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.60019 + 0.383768i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.60019 + 0.383768i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.142 + 0.989i)T \) |
| 7 | \( 1 + (-0.841 - 0.540i)T \) |
| 23 | \( 1 + (4.74 - 0.709i)T \) |
good | 3 | \( 1 + (-0.744 - 1.62i)T + (-1.96 + 2.26i)T^{2} \) |
| 5 | \( 1 + (-1.17 - 1.35i)T + (-0.711 + 4.94i)T^{2} \) |
| 11 | \( 1 + (-0.146 - 1.02i)T + (-10.5 + 3.09i)T^{2} \) |
| 13 | \( 1 + (5.05 - 3.24i)T + (5.40 - 11.8i)T^{2} \) |
| 17 | \( 1 + (-3.82 + 1.12i)T + (14.3 - 9.19i)T^{2} \) |
| 19 | \( 1 + (-5.25 - 1.54i)T + (15.9 + 10.2i)T^{2} \) |
| 29 | \( 1 + (-1.68 + 0.493i)T + (24.3 - 15.6i)T^{2} \) |
| 31 | \( 1 + (-3.38 + 7.40i)T + (-20.3 - 23.4i)T^{2} \) |
| 37 | \( 1 + (4.66 - 5.37i)T + (-5.26 - 36.6i)T^{2} \) |
| 41 | \( 1 + (5.81 + 6.71i)T + (-5.83 + 40.5i)T^{2} \) |
| 43 | \( 1 + (2.39 + 5.24i)T + (-28.1 + 32.4i)T^{2} \) |
| 47 | \( 1 - 3.84T + 47T^{2} \) |
| 53 | \( 1 + (6.93 + 4.45i)T + (22.0 + 48.2i)T^{2} \) |
| 59 | \( 1 + (0.0787 - 0.0505i)T + (24.5 - 53.6i)T^{2} \) |
| 61 | \( 1 + (-2.30 + 5.05i)T + (-39.9 - 46.1i)T^{2} \) |
| 67 | \( 1 + (-1.33 + 9.26i)T + (-64.2 - 18.8i)T^{2} \) |
| 71 | \( 1 + (1.50 - 10.4i)T + (-68.1 - 20.0i)T^{2} \) |
| 73 | \( 1 + (6.45 + 1.89i)T + (61.4 + 39.4i)T^{2} \) |
| 79 | \( 1 + (-9.93 + 6.38i)T + (32.8 - 71.8i)T^{2} \) |
| 83 | \( 1 + (0.824 - 0.951i)T + (-11.8 - 82.1i)T^{2} \) |
| 89 | \( 1 + (-1.83 - 4.01i)T + (-58.2 + 67.2i)T^{2} \) |
| 97 | \( 1 + (9.65 + 11.1i)T + (-13.8 + 96.0i)T^{2} \) |
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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
−11.90368939630019122089606618592, −10.45830436191626134275025948249, −9.792492126743729591879493715091, −9.490603301743342258896528015434, −8.101179002293562843694701948003, −6.87414490472725504990431833446, −5.40641200734786906673176999600, −4.42726646878957797124154456730, −3.27061985130844108280349657647, −2.09418766456390677797262835407,
1.30374291748705932224807578410, 2.99204886343711381461493411472, 4.84708186223958825105309349226, 5.59422543774489176428707996722, 6.93741880632329768729926413814, 7.72969760074897197577607160916, 8.351482196572861019490885379508, 9.554478857368810802016898992524, 10.36510228260671445977972128079, 12.04931968468714550877194940122