| L(s) = 1 | + (−0.133 − 1.40i)2-s + (−1.96 + 0.376i)4-s + (2.47 + 0.901i)5-s + (−2.55 − 0.451i)7-s + (0.793 + 2.71i)8-s + (0.937 − 3.60i)10-s + (−0.556 − 1.52i)11-s + (−1.88 − 2.24i)13-s + (−0.292 + 3.66i)14-s + (3.71 − 1.48i)16-s + (3.28 − 1.89i)17-s + (4.30 − 7.46i)19-s + (−5.20 − 0.837i)20-s + (−2.07 + 0.988i)22-s + (−1.07 − 6.06i)23-s + ⋯ |
| L(s) = 1 | + (−0.0946 − 0.995i)2-s + (−0.982 + 0.188i)4-s + (1.10 + 0.403i)5-s + (−0.967 − 0.170i)7-s + (0.280 + 0.959i)8-s + (0.296 − 1.14i)10-s + (−0.167 − 0.461i)11-s + (−0.522 − 0.622i)13-s + (−0.0782 + 0.979i)14-s + (0.928 − 0.370i)16-s + (0.797 − 0.460i)17-s + (0.988 − 1.71i)19-s + (−1.16 − 0.187i)20-s + (−0.443 + 0.210i)22-s + (−0.223 − 1.26i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 648 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.457 + 0.889i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 648 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.457 + 0.889i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.669174 - 1.09725i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.669174 - 1.09725i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.133 + 1.40i)T \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 + (-2.47 - 0.901i)T + (3.83 + 3.21i)T^{2} \) |
| 7 | \( 1 + (2.55 + 0.451i)T + (6.57 + 2.39i)T^{2} \) |
| 11 | \( 1 + (0.556 + 1.52i)T + (-8.42 + 7.07i)T^{2} \) |
| 13 | \( 1 + (1.88 + 2.24i)T + (-2.25 + 12.8i)T^{2} \) |
| 17 | \( 1 + (-3.28 + 1.89i)T + (8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (-4.30 + 7.46i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (1.07 + 6.06i)T + (-21.6 + 7.86i)T^{2} \) |
| 29 | \( 1 + (-3.88 - 3.25i)T + (5.03 + 28.5i)T^{2} \) |
| 31 | \( 1 + (-3.57 + 0.630i)T + (29.1 - 10.6i)T^{2} \) |
| 37 | \( 1 + (-6.40 + 3.69i)T + (18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (-4.43 - 5.28i)T + (-7.11 + 40.3i)T^{2} \) |
| 43 | \( 1 + (3.77 - 1.37i)T + (32.9 - 27.6i)T^{2} \) |
| 47 | \( 1 + (-0.253 + 1.43i)T + (-44.1 - 16.0i)T^{2} \) |
| 53 | \( 1 - 0.180T + 53T^{2} \) |
| 59 | \( 1 + (0.253 - 0.695i)T + (-45.1 - 37.9i)T^{2} \) |
| 61 | \( 1 + (11.1 + 1.96i)T + (57.3 + 20.8i)T^{2} \) |
| 67 | \( 1 + (10.5 - 8.82i)T + (11.6 - 65.9i)T^{2} \) |
| 71 | \( 1 + (-2.57 - 4.45i)T + (-35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + (2.62 - 4.55i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (-6.72 + 8.01i)T + (-13.7 - 77.7i)T^{2} \) |
| 83 | \( 1 + (7.39 - 8.81i)T + (-14.4 - 81.7i)T^{2} \) |
| 89 | \( 1 + (-0.211 - 0.121i)T + (44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (-1.95 + 0.711i)T + (74.3 - 62.3i)T^{2} \) |
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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
−10.14684246254982785822781442108, −9.715648332384020905724080806464, −8.950207000701637943108435824835, −7.75025986553911507018180374253, −6.60020322303380183883191270587, −5.62370635521767591456700386977, −4.64499349611326132887214926401, −2.99134692854432580885782031178, −2.69737209203866228766939666248, −0.77315455544331474608612190197,
1.55628452629182938158230391681, 3.35439591133692447693529445733, 4.65625303091505312555757972773, 5.81361988427598376015847545376, 6.07221350904163343952733872513, 7.31641582145113929091063261146, 8.084556075649407855247298312676, 9.349535769862806958000615404512, 9.725432880368758724867549378984, 10.21130241839456080063182653574
