| L(s) = 1 | + (0.0955 − 1.41i)2-s + (−1.83 + 0.924i)3-s + (−1.98 − 0.269i)4-s + (0.688 + 1.55i)5-s + (1.12 + 2.67i)6-s + (−2.57 − 1.54i)7-s + (−0.569 + 2.77i)8-s + (0.730 − 0.984i)9-s + (2.25 − 0.823i)10-s + (0.212 + 0.165i)11-s + (3.88 − 1.33i)12-s + (4.07 − 3.88i)13-s + (−2.42 + 3.48i)14-s + (−2.70 − 2.21i)15-s + (3.85 + 1.06i)16-s + (2.14 + 2.60i)17-s + ⋯ |
| L(s) = 1 | + (0.0675 − 0.997i)2-s + (−1.06 + 0.533i)3-s + (−0.990 − 0.134i)4-s + (0.307 + 0.694i)5-s + (0.460 + 1.09i)6-s + (−0.973 − 0.583i)7-s + (−0.201 + 0.979i)8-s + (0.243 − 0.328i)9-s + (0.713 − 0.260i)10-s + (0.0640 + 0.0500i)11-s + (1.12 − 0.385i)12-s + (1.13 − 1.07i)13-s + (−0.647 + 0.931i)14-s + (−0.697 − 0.572i)15-s + (0.963 + 0.267i)16-s + (0.519 + 0.632i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 512 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.163 + 0.986i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 512 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.163 + 0.986i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.617835 - 0.523727i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.617835 - 0.523727i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.0955 + 1.41i)T \) |
| good | 3 | \( 1 + (1.83 - 0.924i)T + (1.78 - 2.40i)T^{2} \) |
| 5 | \( 1 + (-0.688 - 1.55i)T + (-3.35 + 3.70i)T^{2} \) |
| 7 | \( 1 + (2.57 + 1.54i)T + (3.29 + 6.17i)T^{2} \) |
| 11 | \( 1 + (-0.212 - 0.165i)T + (2.67 + 10.6i)T^{2} \) |
| 13 | \( 1 + (-4.07 + 3.88i)T + (0.637 - 12.9i)T^{2} \) |
| 17 | \( 1 + (-2.14 - 2.60i)T + (-3.31 + 16.6i)T^{2} \) |
| 19 | \( 1 + (1.13 + 6.55i)T + (-17.8 + 6.40i)T^{2} \) |
| 23 | \( 1 + (-4.34 - 2.05i)T + (14.5 + 17.7i)T^{2} \) |
| 29 | \( 1 + (1.31 + 0.743i)T + (14.9 + 24.8i)T^{2} \) |
| 31 | \( 1 + (-5.50 + 1.09i)T + (28.6 - 11.8i)T^{2} \) |
| 37 | \( 1 + (2.10 + 1.32i)T + (15.8 + 33.4i)T^{2} \) |
| 41 | \( 1 + (-0.444 - 0.490i)T + (-4.01 + 40.8i)T^{2} \) |
| 43 | \( 1 + (-0.976 - 2.95i)T + (-34.5 + 25.6i)T^{2} \) |
| 47 | \( 1 + (3.59 + 1.92i)T + (26.1 + 39.0i)T^{2} \) |
| 53 | \( 1 + (-5.43 + 3.07i)T + (27.2 - 45.4i)T^{2} \) |
| 59 | \( 1 + (1.97 - 2.07i)T + (-2.89 - 58.9i)T^{2} \) |
| 61 | \( 1 + (-0.133 - 1.80i)T + (-60.3 + 8.95i)T^{2} \) |
| 67 | \( 1 + (4.04 + 3.48i)T + (9.83 + 66.2i)T^{2} \) |
| 71 | \( 1 + (-10.0 - 1.49i)T + (67.9 + 20.6i)T^{2} \) |
| 73 | \( 1 + (-9.12 + 5.46i)T + (34.4 - 64.3i)T^{2} \) |
| 79 | \( 1 + (3.78 + 1.14i)T + (65.6 + 43.8i)T^{2} \) |
| 83 | \( 1 + (-11.0 + 7.01i)T + (35.4 - 75.0i)T^{2} \) |
| 89 | \( 1 + (14.0 - 6.62i)T + (56.4 - 68.7i)T^{2} \) |
| 97 | \( 1 + (-13.1 - 8.79i)T + (37.1 + 89.6i)T^{2} \) |
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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
−10.72327113543354146008642315889, −10.24095625617526397372863021492, −9.394663403685803155898290944551, −8.214929745430241101132124777761, −6.70233909792070394664165072431, −5.91252252147414503341450753819, −4.93082545035924709258089121687, −3.72403307272032102160596523636, −2.82858451445375110590536913999, −0.69162146036487483525176081825,
1.12150907683583564695670852120, 3.52160590798380118252694174207, 4.90380841518932941601128923608, 5.82123684003163273447265819859, 6.32231578128938606159955595190, 7.09797867320173969048013954369, 8.481163330623668406158482121501, 9.106775620857923893083264875212, 9.970633155023644854462359848497, 11.26888342742870303755244191375
