L(s) = 1 | + (0.5 + 0.866i)2-s + (0.5 + 0.866i)3-s + (−0.499 + 0.866i)4-s − 2.13·5-s + (−0.499 + 0.866i)6-s + (−0.0244 + 0.0423i)7-s − 0.999·8-s + (−0.499 + 0.866i)9-s + (−1.06 − 1.85i)10-s + (3.14 + 5.45i)11-s − 0.999·12-s − 0.0489·14-s + (−1.06 − 1.85i)15-s + (−0.5 − 0.866i)16-s + (1.44 − 2.50i)17-s − 0.999·18-s + ⋯ |
L(s) = 1 | + (0.353 + 0.612i)2-s + (0.288 + 0.499i)3-s + (−0.249 + 0.433i)4-s − 0.955·5-s + (−0.204 + 0.353i)6-s + (−0.00924 + 0.0160i)7-s − 0.353·8-s + (−0.166 + 0.288i)9-s + (−0.337 − 0.585i)10-s + (0.949 + 1.64i)11-s − 0.288·12-s − 0.0130·14-s + (−0.275 − 0.477i)15-s + (−0.125 − 0.216i)16-s + (0.350 − 0.607i)17-s − 0.235·18-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1014 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.990 + 0.134i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1014 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.990 + 0.134i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.161446294\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.161446294\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.5 - 0.866i)T \) |
| 3 | \( 1 + (-0.5 - 0.866i)T \) |
| 13 | \( 1 \) |
good | 5 | \( 1 + 2.13T + 5T^{2} \) |
| 7 | \( 1 + (0.0244 - 0.0423i)T + (-3.5 - 6.06i)T^{2} \) |
| 11 | \( 1 + (-3.14 - 5.45i)T + (-5.5 + 9.52i)T^{2} \) |
| 17 | \( 1 + (-1.44 + 2.50i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (3.60 - 6.24i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (1.35 + 2.35i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (2.45 + 4.25i)T + (-14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + 9.00T + 31T^{2} \) |
| 37 | \( 1 + (0.0881 + 0.152i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (4.29 + 7.44i)T + (-20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (3.35 - 5.81i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + 7.20T + 47T^{2} \) |
| 53 | \( 1 - 9.34T + 53T^{2} \) |
| 59 | \( 1 + (2.13 - 3.69i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (3.55 - 6.15i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (-2.69 - 4.66i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (4.35 - 7.54i)T + (-35.5 - 61.4i)T^{2} \) |
| 73 | \( 1 - 14.9T + 73T^{2} \) |
| 79 | \( 1 - 13.8T + 79T^{2} \) |
| 83 | \( 1 - 11.1T + 83T^{2} \) |
| 89 | \( 1 + (-1.96 - 3.39i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (1.23 - 2.14i)T + (-48.5 - 84.0i)T^{2} \) |
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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
−10.17998362539471517611033401387, −9.508440326301406095210207778094, −8.627823122695640906023782938203, −7.73417507662166089131767782665, −7.20652408657618803515130410204, −6.16950732738255903109788941889, −5.03915656967275562129261293691, −4.07071287945415083146305297097, −3.74007661928855701118812023176, −2.06493628089494876027843300420,
0.44741985832714980520524067479, 1.86821030736969631949843729010, 3.46427778106207052608353915587, 3.64996852849227812817929397529, 5.04869103132912338887535370086, 6.14214659458204118387614429504, 6.93090482708190216726784320357, 8.026694910859277286562516148177, 8.705676288825512239072287658961, 9.361884223350330692580462276775