| L(s) = 1 | + (−2.99 + 0.195i)3-s + (−2.50 − 6.87i)5-s + (1.62 + 9.22i)7-s + (8.92 − 1.16i)9-s + (4.02 − 11.0i)11-s + (17.4 + 14.6i)13-s + (8.82 + 20.0i)15-s + (−13.5 − 7.81i)17-s + (−9.08 − 15.7i)19-s + (−6.67 − 27.3i)21-s + (−23.0 − 4.06i)23-s + (−21.8 + 18.3i)25-s + (−26.4 + 5.24i)27-s + (−24.8 − 29.6i)29-s + (4.47 − 25.3i)31-s + ⋯ |
| L(s) = 1 | + (−0.997 + 0.0651i)3-s + (−0.500 − 1.37i)5-s + (0.232 + 1.31i)7-s + (0.991 − 0.129i)9-s + (0.366 − 1.00i)11-s + (1.34 + 1.12i)13-s + (0.588 + 1.33i)15-s + (−0.795 − 0.459i)17-s + (−0.478 − 0.828i)19-s + (−0.317 − 1.30i)21-s + (−1.00 − 0.176i)23-s + (−0.872 + 0.732i)25-s + (−0.980 + 0.194i)27-s + (−0.858 − 1.02i)29-s + (0.144 − 0.818i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 432 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.712 + 0.701i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 432 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.712 + 0.701i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.234205 - 0.572006i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.234205 - 0.572006i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (2.99 - 0.195i)T \) |
| good | 5 | \( 1 + (2.50 + 6.87i)T + (-19.1 + 16.0i)T^{2} \) |
| 7 | \( 1 + (-1.62 - 9.22i)T + (-46.0 + 16.7i)T^{2} \) |
| 11 | \( 1 + (-4.02 + 11.0i)T + (-92.6 - 77.7i)T^{2} \) |
| 13 | \( 1 + (-17.4 - 14.6i)T + (29.3 + 166. i)T^{2} \) |
| 17 | \( 1 + (13.5 + 7.81i)T + (144.5 + 250. i)T^{2} \) |
| 19 | \( 1 + (9.08 + 15.7i)T + (-180.5 + 312. i)T^{2} \) |
| 23 | \( 1 + (23.0 + 4.06i)T + (497. + 180. i)T^{2} \) |
| 29 | \( 1 + (24.8 + 29.6i)T + (-146. + 828. i)T^{2} \) |
| 31 | \( 1 + (-4.47 + 25.3i)T + (-903. - 328. i)T^{2} \) |
| 37 | \( 1 + (-1.45 + 2.52i)T + (-684.5 - 1.18e3i)T^{2} \) |
| 41 | \( 1 + (26.1 - 31.1i)T + (-291. - 1.65e3i)T^{2} \) |
| 43 | \( 1 + (35.7 + 12.9i)T + (1.41e3 + 1.18e3i)T^{2} \) |
| 47 | \( 1 + (-18.5 + 3.26i)T + (2.07e3 - 755. i)T^{2} \) |
| 53 | \( 1 - 12.3iT - 2.80e3T^{2} \) |
| 59 | \( 1 + (4.38 + 12.0i)T + (-2.66e3 + 2.23e3i)T^{2} \) |
| 61 | \( 1 + (15.6 + 88.8i)T + (-3.49e3 + 1.27e3i)T^{2} \) |
| 67 | \( 1 + (-6.68 - 5.61i)T + (779. + 4.42e3i)T^{2} \) |
| 71 | \( 1 + (81.5 + 47.0i)T + (2.52e3 + 4.36e3i)T^{2} \) |
| 73 | \( 1 + (25.3 + 43.8i)T + (-2.66e3 + 4.61e3i)T^{2} \) |
| 79 | \( 1 + (-78.4 + 65.8i)T + (1.08e3 - 6.14e3i)T^{2} \) |
| 83 | \( 1 + (-12.1 - 14.4i)T + (-1.19e3 + 6.78e3i)T^{2} \) |
| 89 | \( 1 + (52.8 - 30.5i)T + (3.96e3 - 6.85e3i)T^{2} \) |
| 97 | \( 1 + (139. + 50.8i)T + (7.20e3 + 6.04e3i)T^{2} \) |
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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
−11.05186002187424318288924475186, −9.392439004498360224027648562090, −8.861055403244980993044247594995, −8.120818104180003361418102609172, −6.46185999420141253841054608468, −5.84690177447930697247402354848, −4.78754525192524928570027624987, −3.98683962511290540504228416305, −1.80641871103046685232122764631, −0.31154970354680367733097692647,
1.52190402551203143235647456922, 3.59685571880993621168272811268, 4.24239348208222187640289728523, 5.77976759393563903172744858496, 6.77703055064783383118386735777, 7.25890965175246915782206602658, 8.262971992413524611327440282158, 10.05489979452948012577812357564, 10.62179142287450137465965640369, 10.95438320412290793830867629373
