L(s) = 1 | + (−0.5 − 0.866i)3-s + (0.5 − 0.866i)5-s + (1 − 1.73i)9-s + (−1.5 − 2.59i)11-s − 13-s − 0.999·15-s + (−2.5 − 4.33i)17-s + (−3 + 5.19i)19-s + (−0.499 − 0.866i)25-s − 5·27-s − 5·29-s + (1 + 1.73i)31-s + (−1.5 + 2.59i)33-s + (2 − 3.46i)37-s + (0.5 + 0.866i)39-s + ⋯ |
L(s) = 1 | + (−0.288 − 0.499i)3-s + (0.223 − 0.387i)5-s + (0.333 − 0.577i)9-s + (−0.452 − 0.783i)11-s − 0.277·13-s − 0.258·15-s + (−0.606 − 1.05i)17-s + (−0.688 + 1.19i)19-s + (−0.0999 − 0.173i)25-s − 0.962·27-s − 0.928·29-s + (0.179 + 0.311i)31-s + (−0.261 + 0.452i)33-s + (0.328 − 0.569i)37-s + (0.0800 + 0.138i)39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.991 - 0.126i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1960 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.991 - 0.126i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.6646231978\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.6646231978\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 + (-0.5 + 0.866i)T \) |
| 7 | \( 1 \) |
good | 3 | \( 1 + (0.5 + 0.866i)T + (-1.5 + 2.59i)T^{2} \) |
| 11 | \( 1 + (1.5 + 2.59i)T + (-5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + T + 13T^{2} \) |
| 17 | \( 1 + (2.5 + 4.33i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (3 - 5.19i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + 5T + 29T^{2} \) |
| 31 | \( 1 + (-1 - 1.73i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (-2 + 3.46i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 - 2T + 41T^{2} \) |
| 43 | \( 1 - 10T + 43T^{2} \) |
| 47 | \( 1 + (4.5 - 7.79i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (3 + 5.19i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (3 + 5.19i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (6 - 10.3i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (-1 - 1.73i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + 71T^{2} \) |
| 73 | \( 1 + (7 + 12.1i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (0.5 - 0.866i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + 12T + 83T^{2} \) |
| 89 | \( 1 + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 - 9T + 97T^{2} \) |
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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
−8.873082516822341898584740381467, −7.86291934283764993665309411163, −7.27601290637810060246760708892, −6.23309982014631108005216564193, −5.79183034374521121195258338485, −4.73695650391501476229455722819, −3.81345955315236696091585750899, −2.65422559635018364511904999340, −1.48148490884598625019512071097, −0.23881461361178018659664430315,
1.85752227474570253921362014755, 2.66349300070233139355651402306, 4.07550742387147876723865384463, 4.63743975748303804781087023258, 5.52139184562914501173299934685, 6.42461629134750665337277215792, 7.24602638786825232450904286351, 7.930501597722165240215229983068, 8.947384973896726967330054325812, 9.690182639471531676258020929056