| L(s) = 1 | + (−1.39 − 0.251i)2-s + (0.0980 − 0.995i)3-s + (1.87 + 0.700i)4-s + (−2.59 + 1.38i)5-s + (−0.386 + 1.36i)6-s + (0.265 + 1.33i)7-s + (−2.43 − 1.44i)8-s + (−0.980 − 0.195i)9-s + (3.95 − 1.27i)10-s + (0.533 − 0.650i)11-s + (0.880 − 1.79i)12-s + (3.22 − 6.04i)13-s + (−0.0337 − 1.92i)14-s + (1.12 + 2.71i)15-s + (3.01 + 2.62i)16-s + (0.823 − 1.98i)17-s + ⋯ |
| L(s) = 1 | + (−0.984 − 0.177i)2-s + (0.0565 − 0.574i)3-s + (0.936 + 0.350i)4-s + (−1.16 + 0.620i)5-s + (−0.157 + 0.555i)6-s + (0.100 + 0.504i)7-s + (−0.859 − 0.511i)8-s + (−0.326 − 0.0650i)9-s + (1.25 − 0.403i)10-s + (0.160 − 0.196i)11-s + (0.254 − 0.518i)12-s + (0.895 − 1.67i)13-s + (−0.00902 − 0.513i)14-s + (0.290 + 0.701i)15-s + (0.754 + 0.655i)16-s + (0.199 − 0.482i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0243 + 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0243 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.445143 - 0.456105i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.445143 - 0.456105i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (1.39 + 0.251i)T \) |
| 3 | \( 1 + (-0.0980 + 0.995i)T \) |
| good | 5 | \( 1 + (2.59 - 1.38i)T + (2.77 - 4.15i)T^{2} \) |
| 7 | \( 1 + (-0.265 - 1.33i)T + (-6.46 + 2.67i)T^{2} \) |
| 11 | \( 1 + (-0.533 + 0.650i)T + (-2.14 - 10.7i)T^{2} \) |
| 13 | \( 1 + (-3.22 + 6.04i)T + (-7.22 - 10.8i)T^{2} \) |
| 17 | \( 1 + (-0.823 + 1.98i)T + (-12.0 - 12.0i)T^{2} \) |
| 19 | \( 1 + (1.67 + 5.50i)T + (-15.7 + 10.5i)T^{2} \) |
| 23 | \( 1 + (-4.34 + 2.90i)T + (8.80 - 21.2i)T^{2} \) |
| 29 | \( 1 + (-2.85 + 2.34i)T + (5.65 - 28.4i)T^{2} \) |
| 31 | \( 1 + (6.18 - 6.18i)T - 31iT^{2} \) |
| 37 | \( 1 + (-2.51 - 0.763i)T + (30.7 + 20.5i)T^{2} \) |
| 41 | \( 1 + (6.09 + 9.11i)T + (-15.6 + 37.8i)T^{2} \) |
| 43 | \( 1 + (0.874 + 8.87i)T + (-42.1 + 8.38i)T^{2} \) |
| 47 | \( 1 + (-0.833 - 0.345i)T + (33.2 + 33.2i)T^{2} \) |
| 53 | \( 1 + (-10.5 - 8.69i)T + (10.3 + 51.9i)T^{2} \) |
| 59 | \( 1 + (4.50 + 8.41i)T + (-32.7 + 49.0i)T^{2} \) |
| 61 | \( 1 + (-3.34 - 0.329i)T + (59.8 + 11.9i)T^{2} \) |
| 67 | \( 1 + (3.83 + 0.377i)T + (65.7 + 13.0i)T^{2} \) |
| 71 | \( 1 + (4.72 - 0.939i)T + (65.5 - 27.1i)T^{2} \) |
| 73 | \( 1 + (0.914 - 4.59i)T + (-67.4 - 27.9i)T^{2} \) |
| 79 | \( 1 + (5.80 - 2.40i)T + (55.8 - 55.8i)T^{2} \) |
| 83 | \( 1 + (-1.84 + 0.561i)T + (69.0 - 46.1i)T^{2} \) |
| 89 | \( 1 + (-8.85 - 5.91i)T + (34.0 + 82.2i)T^{2} \) |
| 97 | \( 1 + (-1.57 + 1.57i)T - 97iT^{2} \) |
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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
−10.94794414305917007936461375155, −10.52680017690761409726782585870, −8.879543823505703796962811821805, −8.447094990471634689332827030111, −7.41612426773214682214995197675, −6.82375593945279055750919354208, −5.52912499361273516116959404029, −3.51495401330073110704362096551, −2.65574074518746324716169463447, −0.62700670129362377235232801495,
1.46307495703907894200758864773, 3.62491051767282389157240104535, 4.46421350549340894040753123144, 6.00485377212718059919870281903, 7.13867315392733335815466550721, 8.075606530115496656410735059920, 8.773662436714786068817234189573, 9.584668204553090778237041668570, 10.63173539961240928030960114588, 11.45828976230514280375196883299
