L(s) = 1 | + (−0.866 − 0.5i)2-s + (−0.866 − 0.5i)3-s + (0.499 + 0.866i)4-s + (0.499 + 0.866i)6-s + (−1.73 + i)7-s − 0.999i·8-s + (0.499 + 0.866i)9-s + (−3 + 5.19i)11-s − 0.999i·12-s + (0.866 − 3.5i)13-s + 1.99·14-s + (−0.5 + 0.866i)16-s + (2.59 − 1.5i)17-s − 0.999i·18-s + (1 + 1.73i)19-s + ⋯ |
L(s) = 1 | + (−0.612 − 0.353i)2-s + (−0.499 − 0.288i)3-s + (0.249 + 0.433i)4-s + (0.204 + 0.353i)6-s + (−0.654 + 0.377i)7-s − 0.353i·8-s + (0.166 + 0.288i)9-s + (−0.904 + 1.56i)11-s − 0.288i·12-s + (0.240 − 0.970i)13-s + 0.534·14-s + (−0.125 + 0.216i)16-s + (0.630 − 0.363i)17-s − 0.235i·18-s + (0.229 + 0.397i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.998 - 0.0471i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.998 - 0.0471i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.05045345576\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.05045345576\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.866 + 0.5i)T \) |
| 3 | \( 1 + (0.866 + 0.5i)T \) |
| 5 | \( 1 \) |
| 13 | \( 1 + (-0.866 + 3.5i)T \) |
good | 7 | \( 1 + (1.73 - i)T + (3.5 - 6.06i)T^{2} \) |
| 11 | \( 1 + (3 - 5.19i)T + (-5.5 - 9.52i)T^{2} \) |
| 17 | \( 1 + (-2.59 + 1.5i)T + (8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (-1 - 1.73i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (-5.19 - 3i)T + (11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (-1.5 + 2.59i)T + (-14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + 4T + 31T^{2} \) |
| 37 | \( 1 + (6.06 + 3.5i)T + (18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (-1.5 + 2.59i)T + (-20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (8.66 - 5i)T + (21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + 6iT - 47T^{2} \) |
| 53 | \( 1 - 3iT - 53T^{2} \) |
| 59 | \( 1 + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (-3.5 - 6.06i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (8.66 + 5i)T + (33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (3 + 5.19i)T + (-35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + 13iT - 73T^{2} \) |
| 79 | \( 1 - 4T + 79T^{2} \) |
| 83 | \( 1 + 6iT - 83T^{2} \) |
| 89 | \( 1 + (-9 + 15.5i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (12.1 - 7i)T + (48.5 - 84.0i)T^{2} \) |
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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
−8.956389062605979920825132135853, −7.77247860537304341025813124184, −7.47318225496887638661761304297, −6.55659659326360303034072045564, −5.53366151530115960643186963417, −4.91549559466148762156176524879, −3.50065826433935028656385380669, −2.63830693879093670153920312134, −1.51709889824810507236412456790, −0.02657176528901211753247650572,
1.19939536916582005569639051604, 2.88468329137275308821534147878, 3.71564622359059726559688391519, 5.00976794389812729559730883889, 5.65375478358991081435350165736, 6.58976472440141440788746872742, 7.02858719526712778661008496885, 8.216734978317376379523365538067, 8.731399793921088149858889591092, 9.581237643712350732338182330539