| L(s) = 1 | + (−0.337 + 1.37i)2-s + (−1.50 + 2.06i)3-s + (−1.77 − 0.926i)4-s + (0.470 − 2.18i)5-s + (−2.33 − 2.76i)6-s + (2.52 − 2.52i)7-s + (1.87 − 2.12i)8-s + (−1.09 − 3.36i)9-s + (2.84 + 1.38i)10-s + (2.45 + 4.82i)11-s + (4.57 − 2.27i)12-s + (−0.583 − 1.79i)13-s + (2.61 + 4.31i)14-s + (3.81 + 4.25i)15-s + (2.28 + 3.28i)16-s + (0.781 − 4.93i)17-s + ⋯ |
| L(s) = 1 | + (−0.238 + 0.971i)2-s + (−0.867 + 1.19i)3-s + (−0.886 − 0.463i)4-s + (0.210 − 0.977i)5-s + (−0.952 − 1.12i)6-s + (0.953 − 0.953i)7-s + (0.661 − 0.750i)8-s + (−0.364 − 1.12i)9-s + (0.899 + 0.437i)10-s + (0.741 + 1.45i)11-s + (1.32 − 0.656i)12-s + (−0.161 − 0.497i)13-s + (0.698 + 1.15i)14-s + (0.985 + 1.09i)15-s + (0.570 + 0.821i)16-s + (0.189 − 1.19i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.553 - 0.833i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.553 - 0.833i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.853988 + 0.458152i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.853988 + 0.458152i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.337 - 1.37i)T \) |
| 5 | \( 1 + (-0.470 + 2.18i)T \) |
| good | 3 | \( 1 + (1.50 - 2.06i)T + (-0.927 - 2.85i)T^{2} \) |
| 7 | \( 1 + (-2.52 + 2.52i)T - 7iT^{2} \) |
| 11 | \( 1 + (-2.45 - 4.82i)T + (-6.46 + 8.89i)T^{2} \) |
| 13 | \( 1 + (0.583 + 1.79i)T + (-10.5 + 7.64i)T^{2} \) |
| 17 | \( 1 + (-0.781 + 4.93i)T + (-16.1 - 5.25i)T^{2} \) |
| 19 | \( 1 + (-0.781 + 4.93i)T + (-18.0 - 5.87i)T^{2} \) |
| 23 | \( 1 + (-3.24 + 1.65i)T + (13.5 - 18.6i)T^{2} \) |
| 29 | \( 1 + (-0.977 - 6.16i)T + (-27.5 + 8.96i)T^{2} \) |
| 31 | \( 1 + (-1.65 - 2.27i)T + (-9.57 + 29.4i)T^{2} \) |
| 37 | \( 1 + (-2.09 - 6.45i)T + (-29.9 + 21.7i)T^{2} \) |
| 41 | \( 1 + (-1.70 + 0.555i)T + (33.1 - 24.0i)T^{2} \) |
| 43 | \( 1 - 6.56T + 43T^{2} \) |
| 47 | \( 1 + (0.206 + 1.30i)T + (-44.6 + 14.5i)T^{2} \) |
| 53 | \( 1 + (-7.84 + 10.7i)T + (-16.3 - 50.4i)T^{2} \) |
| 59 | \( 1 + (2.90 - 5.69i)T + (-34.6 - 47.7i)T^{2} \) |
| 61 | \( 1 + (3.13 + 6.15i)T + (-35.8 + 49.3i)T^{2} \) |
| 67 | \( 1 + (1.24 - 0.906i)T + (20.7 - 63.7i)T^{2} \) |
| 71 | \( 1 + (-12.4 - 9.02i)T + (21.9 + 67.5i)T^{2} \) |
| 73 | \( 1 + (3.00 + 5.89i)T + (-42.9 + 59.0i)T^{2} \) |
| 79 | \( 1 + (4.66 + 3.39i)T + (24.4 + 75.1i)T^{2} \) |
| 83 | \( 1 + (-4.81 - 6.62i)T + (-25.6 + 78.9i)T^{2} \) |
| 89 | \( 1 + (-3.48 + 10.7i)T + (-72.0 - 52.3i)T^{2} \) |
| 97 | \( 1 + (13.9 - 2.21i)T + (92.2 - 29.9i)T^{2} \) |
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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
−11.19891916893996308760407520436, −10.25191896267835831450741536118, −9.588238248186001636467955636773, −8.852728218423156756939002942693, −7.57676590811523388856482823318, −6.75248423601674602071634098769, −5.14116076517773588344303651001, −4.90415943956543351556794792654, −4.19485481887081804990666914077, −0.966656110630154127059756215134,
1.32422695914009059701819831666, 2.38503041584317574871425065396, 3.87741456106216344477278532760, 5.69939271081612228697774572025, 6.10846558456620524578342649382, 7.55709232075754357446504908786, 8.318740140350792437240848565264, 9.375786641102530965288903179252, 10.73875473837308030077846301058, 11.25672102683070631516045774810
