| L(s) = 1 | + (−1.49 − 1.77i)2-s + (−0.601 + 1.62i)3-s + (−0.587 + 3.32i)4-s + (−1.83 + 0.667i)5-s + (3.78 − 1.35i)6-s + (1.75 − 1.97i)7-s + (2.77 − 1.60i)8-s + (−2.27 − 1.95i)9-s + (3.91 + 2.26i)10-s + (1.80 − 4.95i)11-s + (−5.05 − 2.95i)12-s + (2.93 − 3.49i)13-s + (−6.13 − 0.169i)14-s + (0.0196 − 3.37i)15-s + (−0.627 − 0.228i)16-s + (0.614 − 1.06i)17-s + ⋯ |
| L(s) = 1 | + (−1.05 − 1.25i)2-s + (−0.347 + 0.937i)3-s + (−0.293 + 1.66i)4-s + (−0.819 + 0.298i)5-s + (1.54 − 0.552i)6-s + (0.663 − 0.748i)7-s + (0.980 − 0.566i)8-s + (−0.758 − 0.651i)9-s + (1.23 + 0.715i)10-s + (0.543 − 1.49i)11-s + (−1.45 − 0.853i)12-s + (0.813 − 0.969i)13-s + (−1.63 − 0.0452i)14-s + (0.00506 − 0.872i)15-s + (−0.156 − 0.0570i)16-s + (0.148 − 0.257i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.329 + 0.944i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.329 + 0.944i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.291270 - 0.410186i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.291270 - 0.410186i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (0.601 - 1.62i)T \) |
| 7 | \( 1 + (-1.75 + 1.97i)T \) |
| good | 2 | \( 1 + (1.49 + 1.77i)T + (-0.347 + 1.96i)T^{2} \) |
| 5 | \( 1 + (1.83 - 0.667i)T + (3.83 - 3.21i)T^{2} \) |
| 11 | \( 1 + (-1.80 + 4.95i)T + (-8.42 - 7.07i)T^{2} \) |
| 13 | \( 1 + (-2.93 + 3.49i)T + (-2.25 - 12.8i)T^{2} \) |
| 17 | \( 1 + (-0.614 + 1.06i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (1.16 - 0.671i)T + (9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (-3.17 - 0.559i)T + (21.6 + 7.86i)T^{2} \) |
| 29 | \( 1 + (0.263 + 0.314i)T + (-5.03 + 28.5i)T^{2} \) |
| 31 | \( 1 + (8.34 + 1.47i)T + (29.1 + 10.6i)T^{2} \) |
| 37 | \( 1 + (-5.01 + 8.68i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (-4.15 - 3.49i)T + (7.11 + 40.3i)T^{2} \) |
| 43 | \( 1 + (-2.21 - 0.806i)T + (32.9 + 27.6i)T^{2} \) |
| 47 | \( 1 + (-0.110 - 0.626i)T + (-44.1 + 16.0i)T^{2} \) |
| 53 | \( 1 + 6.55iT - 53T^{2} \) |
| 59 | \( 1 + (0.369 - 0.134i)T + (45.1 - 37.9i)T^{2} \) |
| 61 | \( 1 + (13.5 - 2.39i)T + (57.3 - 20.8i)T^{2} \) |
| 67 | \( 1 + (0.495 + 0.415i)T + (11.6 + 65.9i)T^{2} \) |
| 71 | \( 1 + (-5.77 - 3.33i)T + (35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + (-11.6 + 6.73i)T + (36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (4.65 - 3.90i)T + (13.7 - 77.7i)T^{2} \) |
| 83 | \( 1 + (12.2 - 10.2i)T + (14.4 - 81.7i)T^{2} \) |
| 89 | \( 1 + (0.131 + 0.228i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (-2.51 + 6.91i)T + (-74.3 - 62.3i)T^{2} \) |
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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
−11.51072257781939371099037766756, −11.04920908657496150003445501015, −10.74058412936095076966358134307, −9.440696556923995302056733116615, −8.543727270970509789104328078037, −7.68967572568939442313908029849, −5.76607396348303753154203318674, −3.93522272151924195841352072304, −3.29375499677629868072393976028, −0.72559567736325321570214497762,
1.60893163911494640621122075574, 4.62250490110895474363694701861, 5.96216663035386077239123224835, 6.95085334259002864982727083533, 7.70537088959740864035547058058, 8.591953501371737066067911864776, 9.319906060251522832447760769364, 10.95587222047800008704050473165, 11.92174136185190947707731559487, 12.64890422503466009570502351717
