| L(s) = 1 | + (−0.466 − 1.33i)2-s + (−0.572 − 0.856i)3-s + (−1.56 + 1.24i)4-s + (−0.690 − 3.47i)5-s + (−0.876 + 1.16i)6-s + (0.983 + 2.37i)7-s + (2.39 + 1.50i)8-s + (0.741 − 1.79i)9-s + (−4.31 + 2.54i)10-s + (1.29 + 0.862i)11-s + (1.96 + 0.626i)12-s + (−0.115 + 0.581i)13-s + (2.71 − 2.42i)14-s + (−2.57 + 2.57i)15-s + (0.893 − 3.89i)16-s + (4.30 + 4.30i)17-s + ⋯ |
| L(s) = 1 | + (−0.330 − 0.943i)2-s + (−0.330 − 0.494i)3-s + (−0.782 + 0.623i)4-s + (−0.308 − 1.55i)5-s + (−0.357 + 0.475i)6-s + (0.371 + 0.897i)7-s + (0.846 + 0.532i)8-s + (0.247 − 0.596i)9-s + (−1.36 + 0.803i)10-s + (0.389 + 0.259i)11-s + (0.566 + 0.180i)12-s + (−0.0320 + 0.161i)13-s + (0.724 − 0.647i)14-s + (−0.665 + 0.665i)15-s + (0.223 − 0.974i)16-s + (1.04 + 1.04i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 64 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.422 + 0.906i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 64 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.422 + 0.906i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.360824 - 0.566187i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.360824 - 0.566187i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.466 + 1.33i)T \) |
| good | 3 | \( 1 + (0.572 + 0.856i)T + (-1.14 + 2.77i)T^{2} \) |
| 5 | \( 1 + (0.690 + 3.47i)T + (-4.61 + 1.91i)T^{2} \) |
| 7 | \( 1 + (-0.983 - 2.37i)T + (-4.94 + 4.94i)T^{2} \) |
| 11 | \( 1 + (-1.29 - 0.862i)T + (4.20 + 10.1i)T^{2} \) |
| 13 | \( 1 + (0.115 - 0.581i)T + (-12.0 - 4.97i)T^{2} \) |
| 17 | \( 1 + (-4.30 - 4.30i)T + 17iT^{2} \) |
| 19 | \( 1 + (-3.59 - 0.714i)T + (17.5 + 7.27i)T^{2} \) |
| 23 | \( 1 + (7.79 + 3.22i)T + (16.2 + 16.2i)T^{2} \) |
| 29 | \( 1 + (0.373 - 0.249i)T + (11.0 - 26.7i)T^{2} \) |
| 31 | \( 1 - 1.08iT - 31T^{2} \) |
| 37 | \( 1 + (4.16 - 0.827i)T + (34.1 - 14.1i)T^{2} \) |
| 41 | \( 1 + (-5.15 - 2.13i)T + (28.9 + 28.9i)T^{2} \) |
| 43 | \( 1 + (1.55 - 2.33i)T + (-16.4 - 39.7i)T^{2} \) |
| 47 | \( 1 + (-6.43 - 6.43i)T + 47iT^{2} \) |
| 53 | \( 1 + (1.22 + 0.816i)T + (20.2 + 48.9i)T^{2} \) |
| 59 | \( 1 + (1.46 + 7.36i)T + (-54.5 + 22.5i)T^{2} \) |
| 61 | \( 1 + (-6.45 - 9.66i)T + (-23.3 + 56.3i)T^{2} \) |
| 67 | \( 1 + (-3.30 - 4.94i)T + (-25.6 + 61.8i)T^{2} \) |
| 71 | \( 1 + (3.75 + 9.05i)T + (-50.2 + 50.2i)T^{2} \) |
| 73 | \( 1 + (1.19 - 2.89i)T + (-51.6 - 51.6i)T^{2} \) |
| 79 | \( 1 + (0.934 - 0.934i)T - 79iT^{2} \) |
| 83 | \( 1 + (16.5 + 3.29i)T + (76.6 + 31.7i)T^{2} \) |
| 89 | \( 1 + (6.15 - 2.54i)T + (62.9 - 62.9i)T^{2} \) |
| 97 | \( 1 + 5.79iT - 97T^{2} \) |
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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
−14.36509749892613311170059015874, −12.82243172816970350457666162817, −12.23232757028910530753480703582, −11.75709139591630080941778012507, −9.878160454733024758793515603322, −8.841065698035065127130540642565, −7.909128723717137043888414454915, −5.63024795968274408728596317075, −4.11921013436627371972990494679, −1.43690184400027397398068668719,
3.89682227061201418896200626190, 5.57145809659843768591239820188, 7.15244427008299414040052389236, 7.74356111803444038464461277673, 9.811496677147068946997176744979, 10.50706328561588648073559924500, 11.54910833052109849413483450952, 13.91965051859017711393862216061, 14.12431705120662205837273658232, 15.47121099174598421302872603816
