| L(s) = 1 | + (−0.142 + 0.989i)2-s + (−1.13 − 2.48i)3-s + (−0.959 − 0.281i)4-s + (−2.36 − 2.72i)5-s + (2.61 − 0.768i)6-s + (0.841 + 0.540i)7-s + (0.415 − 0.909i)8-s + (−2.91 + 3.36i)9-s + (3.03 − 1.95i)10-s + (0.471 + 3.28i)11-s + (0.388 + 2.70i)12-s + (−4.79 + 3.07i)13-s + (−0.654 + 0.755i)14-s + (−4.09 + 8.96i)15-s + (0.841 + 0.540i)16-s + (4.13 − 1.21i)17-s + ⋯ |
| L(s) = 1 | + (−0.100 + 0.699i)2-s + (−0.654 − 1.43i)3-s + (−0.479 − 0.140i)4-s + (−1.05 − 1.22i)5-s + (1.06 − 0.313i)6-s + (0.317 + 0.204i)7-s + (0.146 − 0.321i)8-s + (−0.971 + 1.12i)9-s + (0.960 − 0.617i)10-s + (0.142 + 0.989i)11-s + (0.112 + 0.779i)12-s + (−1.32 + 0.853i)13-s + (−0.175 + 0.201i)14-s + (−1.05 + 2.31i)15-s + (0.210 + 0.135i)16-s + (1.00 − 0.294i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 322 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.974 - 0.223i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 322 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.974 - 0.223i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.0212380 + 0.187834i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0212380 + 0.187834i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.142 - 0.989i)T \) |
| 7 | \( 1 + (-0.841 - 0.540i)T \) |
| 23 | \( 1 + (4.09 + 2.49i)T \) |
| good | 3 | \( 1 + (1.13 + 2.48i)T + (-1.96 + 2.26i)T^{2} \) |
| 5 | \( 1 + (2.36 + 2.72i)T + (-0.711 + 4.94i)T^{2} \) |
| 11 | \( 1 + (-0.471 - 3.28i)T + (-10.5 + 3.09i)T^{2} \) |
| 13 | \( 1 + (4.79 - 3.07i)T + (5.40 - 11.8i)T^{2} \) |
| 17 | \( 1 + (-4.13 + 1.21i)T + (14.3 - 9.19i)T^{2} \) |
| 19 | \( 1 + (4.49 + 1.32i)T + (15.9 + 10.2i)T^{2} \) |
| 29 | \( 1 + (-4.68 + 1.37i)T + (24.3 - 15.6i)T^{2} \) |
| 31 | \( 1 + (-0.704 + 1.54i)T + (-20.3 - 23.4i)T^{2} \) |
| 37 | \( 1 + (5.93 - 6.84i)T + (-5.26 - 36.6i)T^{2} \) |
| 41 | \( 1 + (1.90 + 2.20i)T + (-5.83 + 40.5i)T^{2} \) |
| 43 | \( 1 + (4.01 + 8.78i)T + (-28.1 + 32.4i)T^{2} \) |
| 47 | \( 1 + 1.64T + 47T^{2} \) |
| 53 | \( 1 + (5.65 + 3.63i)T + (22.0 + 48.2i)T^{2} \) |
| 59 | \( 1 + (1.22 - 0.784i)T + (24.5 - 53.6i)T^{2} \) |
| 61 | \( 1 + (3.05 - 6.69i)T + (-39.9 - 46.1i)T^{2} \) |
| 67 | \( 1 + (-1.33 + 9.28i)T + (-64.2 - 18.8i)T^{2} \) |
| 71 | \( 1 + (-0.753 + 5.24i)T + (-68.1 - 20.0i)T^{2} \) |
| 73 | \( 1 + (12.4 + 3.66i)T + (61.4 + 39.4i)T^{2} \) |
| 79 | \( 1 + (8.52 - 5.47i)T + (32.8 - 71.8i)T^{2} \) |
| 83 | \( 1 + (-8.50 + 9.81i)T + (-11.8 - 82.1i)T^{2} \) |
| 89 | \( 1 + (-2.46 - 5.39i)T + (-58.2 + 67.2i)T^{2} \) |
| 97 | \( 1 + (-2.45 - 2.83i)T + (-13.8 + 96.0i)T^{2} \) |
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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
−11.85227013905603724681213935033, −10.08595517655031725443986383896, −8.821845947223125476800651232884, −7.979448588570407615837305162504, −7.32368006662444965326005612571, −6.49181878201208703306459268068, −5.10765915820203967039315291918, −4.45441646989956342934386916604, −1.80256760592422016244198746629, −0.14952928722563176302558192194,
3.05998925885828429452479110432, 3.77935517860034622130618142115, 4.83413194390641225883208593605, 6.02016756610713624004783407215, 7.54561208556546949973029120044, 8.444292150148157926577278881915, 9.957817505137007681614594557391, 10.34989616518173435428062457801, 11.06863830938524943157932529118, 11.70247744678090302186582854019
