L(s) = 1 | + (−0.637 + 1.61i)3-s + (1.68 − 2.92i)5-s + (−2.35 − 4.07i)7-s + (−2.18 − 2.05i)9-s + (−0.437 − 0.758i)11-s + (0.686 − 1.18i)13-s + (3.62 + 4.57i)15-s − 2.37·17-s + 5.57·19-s + (8.05 − 1.18i)21-s + (−2.35 + 4.07i)23-s + (−3.18 − 5.51i)25-s + (4.70 − 2.20i)27-s + (2.68 + 4.65i)29-s + (3.22 − 5.58i)31-s + ⋯ |
L(s) = 1 | + (−0.368 + 0.929i)3-s + (0.754 − 1.30i)5-s + (−0.888 − 1.53i)7-s + (−0.728 − 0.684i)9-s + (−0.131 − 0.228i)11-s + (0.190 − 0.329i)13-s + (0.936 + 1.18i)15-s − 0.575·17-s + 1.27·19-s + (1.75 − 0.259i)21-s + (−0.490 + 0.849i)23-s + (−0.637 − 1.10i)25-s + (0.905 − 0.425i)27-s + (0.498 + 0.863i)29-s + (0.579 − 1.00i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 288 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.450 + 0.892i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 288 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.450 + 0.892i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.899998 - 0.553914i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.899998 - 0.553914i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (0.637 - 1.61i)T \) |
good | 5 | \( 1 + (-1.68 + 2.92i)T + (-2.5 - 4.33i)T^{2} \) |
| 7 | \( 1 + (2.35 + 4.07i)T + (-3.5 + 6.06i)T^{2} \) |
| 11 | \( 1 + (0.437 + 0.758i)T + (-5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + (-0.686 + 1.18i)T + (-6.5 - 11.2i)T^{2} \) |
| 17 | \( 1 + 2.37T + 17T^{2} \) |
| 19 | \( 1 - 5.57T + 19T^{2} \) |
| 23 | \( 1 + (2.35 - 4.07i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (-2.68 - 4.65i)T + (-14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + (-3.22 + 5.58i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 - 4T + 37T^{2} \) |
| 41 | \( 1 + (0.5 - 0.866i)T + (-20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (-0.437 - 0.758i)T + (-21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + (2.35 + 4.07i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + 4T + 53T^{2} \) |
| 59 | \( 1 + (4.26 - 7.38i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (-1.05 - 1.83i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (-4.26 + 7.38i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 - 9.40T + 71T^{2} \) |
| 73 | \( 1 - 10.3T + 73T^{2} \) |
| 79 | \( 1 + (3.22 + 5.58i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (-1.47 - 2.55i)T + (-41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 - 12.7T + 89T^{2} \) |
| 97 | \( 1 + (4.5 + 7.79i)T + (-48.5 + 84.0i)T^{2} \) |
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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
−11.52858417100605802139663582416, −10.47834179185199976473151149605, −9.738982416177387233802943556580, −9.198191997141157078962445602492, −7.895801209949622489886278484385, −6.47263953694936221403749624326, −5.45513769581762674236826534238, −4.48147036015176642210373706610, −3.40274227448249267615359237043, −0.844747079800542815861732081025,
2.23110178459491883231534488872, 2.95163567511660650148391584590, 5.32192992182478983576924522287, 6.38594382169025993340013511890, 6.60509565619710953764130840044, 8.022837797734694856056413815867, 9.233423764321983281558675211584, 10.09767422587562927277019333744, 11.20104438253484637502836080132, 12.01655174523271428784202014192