| L(s) = 1 | + (0.306 − 1.38i)2-s + (0.156 + 0.534i)3-s + (−1.81 − 0.847i)4-s + (1.75 + 1.38i)5-s + (0.785 − 0.0526i)6-s + (−2.85 + 0.409i)7-s + (−1.72 + 2.24i)8-s + (2.26 − 1.45i)9-s + (2.44 − 2.00i)10-s + (1.05 + 0.481i)11-s + (0.168 − 1.10i)12-s + (−0.380 + 2.64i)13-s + (−0.308 + 4.06i)14-s + (−0.461 + 1.15i)15-s + (2.56 + 3.06i)16-s + (3.94 + 4.54i)17-s + ⋯ |
| L(s) = 1 | + (0.216 − 0.976i)2-s + (0.0905 + 0.308i)3-s + (−0.905 − 0.423i)4-s + (0.786 + 0.617i)5-s + (0.320 − 0.0214i)6-s + (−1.07 + 0.154i)7-s + (−0.609 + 0.792i)8-s + (0.754 − 0.484i)9-s + (0.773 − 0.633i)10-s + (0.317 + 0.145i)11-s + (0.0485 − 0.317i)12-s + (−0.105 + 0.734i)13-s + (−0.0825 + 1.08i)14-s + (−0.119 + 0.298i)15-s + (0.641 + 0.767i)16-s + (0.956 + 1.10i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 920 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.998 - 0.0451i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 920 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.998 - 0.0451i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.69486 + 0.0383115i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.69486 + 0.0383115i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.306 + 1.38i)T \) |
| 5 | \( 1 + (-1.75 - 1.38i)T \) |
| 23 | \( 1 + (-0.327 - 4.78i)T \) |
| good | 3 | \( 1 + (-0.156 - 0.534i)T + (-2.52 + 1.62i)T^{2} \) |
| 7 | \( 1 + (2.85 - 0.409i)T + (6.71 - 1.97i)T^{2} \) |
| 11 | \( 1 + (-1.05 - 0.481i)T + (7.20 + 8.31i)T^{2} \) |
| 13 | \( 1 + (0.380 - 2.64i)T + (-12.4 - 3.66i)T^{2} \) |
| 17 | \( 1 + (-3.94 - 4.54i)T + (-2.41 + 16.8i)T^{2} \) |
| 19 | \( 1 + (3.07 + 2.66i)T + (2.70 + 18.8i)T^{2} \) |
| 29 | \( 1 + (1.10 - 0.959i)T + (4.12 - 28.7i)T^{2} \) |
| 31 | \( 1 + (-1.70 + 5.81i)T + (-26.0 - 16.7i)T^{2} \) |
| 37 | \( 1 + (-0.928 - 1.44i)T + (-15.3 + 33.6i)T^{2} \) |
| 41 | \( 1 + (-9.25 - 5.94i)T + (17.0 + 37.2i)T^{2} \) |
| 43 | \( 1 + (-2.75 + 0.809i)T + (36.1 - 23.2i)T^{2} \) |
| 47 | \( 1 - 7.92T + 47T^{2} \) |
| 53 | \( 1 + (4.60 - 0.661i)T + (50.8 - 14.9i)T^{2} \) |
| 59 | \( 1 + (1.84 - 12.8i)T + (-56.6 - 16.6i)T^{2} \) |
| 61 | \( 1 + (-7.19 - 2.11i)T + (51.3 + 32.9i)T^{2} \) |
| 67 | \( 1 + (3.49 + 7.64i)T + (-43.8 + 50.6i)T^{2} \) |
| 71 | \( 1 + (3.54 - 1.62i)T + (46.4 - 53.6i)T^{2} \) |
| 73 | \( 1 + (9.67 + 8.38i)T + (10.3 + 72.2i)T^{2} \) |
| 79 | \( 1 + (-0.919 + 6.39i)T + (-75.7 - 22.2i)T^{2} \) |
| 83 | \( 1 + (-4.80 + 3.09i)T + (34.4 - 75.4i)T^{2} \) |
| 89 | \( 1 + (0.360 + 1.22i)T + (-74.8 + 48.1i)T^{2} \) |
| 97 | \( 1 + (3.88 + 2.49i)T + (40.2 + 88.2i)T^{2} \) |
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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
−10.09376365952379602197352040085, −9.410921866647753723731949306930, −9.078967317616469803923315115599, −7.48918148791398308859211253832, −6.34616414049532519163860137751, −5.85635705078659041729416227975, −4.40782911428430286118175230435, −3.62908871662700362348995263005, −2.67610418463546072139325772259, −1.45400715560722691593457908026,
0.825591267387965537643588229122, 2.68543429697616679372344971136, 3.97246154239133984083250775152, 5.00420378260779718579524060355, 5.84760042548855261396346603308, 6.61462013996103040274717362241, 7.41972711358038007154742604059, 8.287124265947472131468956376628, 9.184736721298271880953148328138, 9.880122559614938045965644488210
