L(s) = 1 | + (0.615 − 1.61i)3-s − 3.91·5-s + (−2.51 + 0.813i)7-s + (−2.24 − 1.99i)9-s − 3.69i·11-s + (−0.480 + 0.277i)13-s + (−2.41 + 6.33i)15-s + (−2.91 − 5.05i)17-s + (4.62 + 2.66i)19-s + (−0.233 + 4.57i)21-s + 2.27i·23-s + 10.3·25-s + (−4.60 + 2.40i)27-s + (3.53 + 2.04i)29-s + (−7.00 − 4.04i)31-s + ⋯ |
L(s) = 1 | + (0.355 − 0.934i)3-s − 1.75·5-s + (−0.951 + 0.307i)7-s + (−0.747 − 0.664i)9-s − 1.11i·11-s + (−0.133 + 0.0769i)13-s + (−0.622 + 1.63i)15-s + (−0.707 − 1.22i)17-s + (1.06 + 0.612i)19-s + (−0.0509 + 0.998i)21-s + 0.474i·23-s + 2.06·25-s + (−0.886 + 0.461i)27-s + (0.656 + 0.379i)29-s + (−1.25 − 0.726i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.945 + 0.325i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.945 + 0.325i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.0867807 - 0.519000i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.0867807 - 0.519000i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (-0.615 + 1.61i)T \) |
| 7 | \( 1 + (2.51 - 0.813i)T \) |
good | 5 | \( 1 + 3.91T + 5T^{2} \) |
| 11 | \( 1 + 3.69iT - 11T^{2} \) |
| 13 | \( 1 + (0.480 - 0.277i)T + (6.5 - 11.2i)T^{2} \) |
| 17 | \( 1 + (2.91 + 5.05i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (-4.62 - 2.66i)T + (9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 - 2.27iT - 23T^{2} \) |
| 29 | \( 1 + (-3.53 - 2.04i)T + (14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + (7.00 + 4.04i)T + (15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (-3.89 + 6.75i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (3.59 + 6.22i)T + (-20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (0.754 - 1.30i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (1.41 + 2.44i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (-0.0415 + 0.0239i)T + (26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (4.45 - 7.71i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (-6.03 + 3.48i)T + (30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (0.587 - 1.01i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + 6.71iT - 71T^{2} \) |
| 73 | \( 1 + (3.52 - 2.03i)T + (36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (-1.97 - 3.41i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (3.84 - 6.66i)T + (-41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 + (-2.71 + 4.69i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (13.9 + 8.07i)T + (48.5 + 84.0i)T^{2} \) |
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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
−11.75026332647696286116702463134, −11.09687999971778897673904711821, −9.369090037121318388675544738332, −8.565626300981847354951859887405, −7.56816905002758238815125870206, −6.93625283994926087100606424708, −5.60615898061557036479574158675, −3.76158741811504722033127454176, −2.94311735877362145157990498035, −0.38968082043150080744755969090,
3.05368711137577362228264469935, 4.01371910914831372318582235316, 4.81541153386929961870645534626, 6.68279325755559618031652214233, 7.67633199591487795566228605466, 8.597521511834171998216621504007, 9.650541639869389037609361969546, 10.53838625042999857871364502132, 11.43867826333379027955915447564, 12.37348356488786808343656818683