| 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
−11.45828976230514280375196883299, −10.63173539961240928030960114588, −9.584668204553090778237041668570, −8.773662436714786068817234189573, −8.075606530115496656410735059920, −7.13867315392733335815466550721, −6.00485377212718059919870281903, −4.46421350549340894040753123144, −3.62491051767282389157240104535, −1.46307495703907894200758864773,
0.62700670129362377235232801495, 2.65574074518746324716169463447, 3.51495401330073110704362096551, 5.52912499361273516116959404029, 6.82375593945279055750919354208, 7.41612426773214682214995197675, 8.447094990471634689332827030111, 8.879543823505703796962811821805, 10.52680017690761409726782585870, 10.94794414305917007936461375155
