L(s) = 1 | + (0.349 + 1.69i)3-s + 3.69·5-s + (1.40 + 2.24i)7-s + (−2.75 + 1.18i)9-s + 1.47·11-s + (−1.34 − 2.33i)13-s + (1.29 + 6.27i)15-s + (3.28 + 5.69i)17-s + (0.444 − 0.769i)19-s + (−3.31 + 3.16i)21-s − 6.28·23-s + 8.68·25-s + (−2.97 − 4.25i)27-s + (1.25 − 2.17i)29-s + (3.40 − 5.89i)31-s + ⋯ |
L(s) = 1 | + (0.201 + 0.979i)3-s + 1.65·5-s + (0.531 + 0.847i)7-s + (−0.918 + 0.395i)9-s + 0.445·11-s + (−0.374 − 0.648i)13-s + (0.334 + 1.62i)15-s + (0.797 + 1.38i)17-s + (0.101 − 0.176i)19-s + (−0.722 + 0.691i)21-s − 1.31·23-s + 1.73·25-s + (−0.572 − 0.819i)27-s + (0.233 − 0.403i)29-s + (0.611 − 1.05i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.190 - 0.981i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.190 - 0.981i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.365170434\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.365170434\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (-0.349 - 1.69i)T \) |
| 7 | \( 1 + (-1.40 - 2.24i)T \) |
good | 5 | \( 1 - 3.69T + 5T^{2} \) |
| 11 | \( 1 - 1.47T + 11T^{2} \) |
| 13 | \( 1 + (1.34 + 2.33i)T + (-6.5 + 11.2i)T^{2} \) |
| 17 | \( 1 + (-3.28 - 5.69i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (-0.444 + 0.769i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + 6.28T + 23T^{2} \) |
| 29 | \( 1 + (-1.25 + 2.17i)T + (-14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + (-3.40 + 5.89i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (1.38 - 2.40i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (2.05 + 3.56i)T + (-20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (0.00618 - 0.0107i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (3.49 + 6.05i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (1.60 + 2.78i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (-3.45 + 5.98i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (-2.86 - 4.96i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (4.73 - 8.19i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 - 5.46T + 71T^{2} \) |
| 73 | \( 1 + (6.03 + 10.4i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (-5.72 - 9.91i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (2.23 - 3.87i)T + (-41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 + (4.43 - 7.68i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (6.58 - 11.4i)T + (-48.5 - 84.0i)T^{2} \) |
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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
−9.956018844603940940271257664526, −9.597820959061696129369322458518, −8.568576385477626517475421844961, −8.043495636901880489024006085301, −6.34462652926123119907121213276, −5.71278249256307805134153042209, −5.16061646378169559277793887487, −3.94543305387240282319054500557, −2.66538709856972608246273600499, −1.80679661295125280428201686411,
1.17688736407831848937463953416, 1.98945166766857644454963637492, 3.12480129779635462651186558067, 4.68701181304165230751247305788, 5.62131860316359037425551068739, 6.50371408377511550642947600548, 7.13536003472815780391527582223, 8.018653958606438185261821160049, 9.048279954618423477633324843195, 9.733864437816586783745216670303