| L(s) = 1 | + (−0.655 − 0.139i)2-s + (0.328 − 3.12i)3-s + (−1.41 − 0.631i)4-s + (−1.43 − 1.59i)5-s + (−0.649 + 1.99i)6-s + (1.92 + 1.39i)8-s + (−6.70 − 1.42i)9-s + (0.717 + 1.24i)10-s + (−2.17 + 2.50i)11-s + (−2.43 + 4.21i)12-s + (−0.781 − 2.40i)13-s + (−5.44 + 3.95i)15-s + (1.01 + 1.12i)16-s + (−1.75 + 0.372i)17-s + (4.19 + 1.86i)18-s + (6.14 − 2.73i)19-s + ⋯ |
| L(s) = 1 | + (−0.463 − 0.0984i)2-s + (0.189 − 1.80i)3-s + (−0.708 − 0.315i)4-s + (−0.641 − 0.712i)5-s + (−0.265 + 0.816i)6-s + (0.680 + 0.494i)8-s + (−2.23 − 0.475i)9-s + (0.227 + 0.393i)10-s + (−0.654 + 0.755i)11-s + (−0.703 + 1.21i)12-s + (−0.216 − 0.666i)13-s + (−1.40 + 1.02i)15-s + (0.252 + 0.280i)16-s + (−0.424 + 0.0903i)17-s + (0.989 + 0.440i)18-s + (1.40 − 0.627i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 539 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.224 - 0.974i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 539 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.224 - 0.974i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.246932 + 0.310368i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.246932 + 0.310368i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 \) |
| 11 | \( 1 + (2.17 - 2.50i)T \) |
| good | 2 | \( 1 + (0.655 + 0.139i)T + (1.82 + 0.813i)T^{2} \) |
| 3 | \( 1 + (-0.328 + 3.12i)T + (-2.93 - 0.623i)T^{2} \) |
| 5 | \( 1 + (1.43 + 1.59i)T + (-0.522 + 4.97i)T^{2} \) |
| 13 | \( 1 + (0.781 + 2.40i)T + (-10.5 + 7.64i)T^{2} \) |
| 17 | \( 1 + (1.75 - 0.372i)T + (15.5 - 6.91i)T^{2} \) |
| 19 | \( 1 + (-6.14 + 2.73i)T + (12.7 - 14.1i)T^{2} \) |
| 23 | \( 1 + (-1.58 + 2.74i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (-0.747 + 0.543i)T + (8.96 - 27.5i)T^{2} \) |
| 31 | \( 1 + (2.00 - 2.23i)T + (-3.24 - 30.8i)T^{2} \) |
| 37 | \( 1 + (-0.157 - 1.49i)T + (-36.1 + 7.69i)T^{2} \) |
| 41 | \( 1 + (-4.49 - 3.26i)T + (12.6 + 38.9i)T^{2} \) |
| 43 | \( 1 + 8.42T + 43T^{2} \) |
| 47 | \( 1 + (4.01 - 1.78i)T + (31.4 - 34.9i)T^{2} \) |
| 53 | \( 1 + (-0.446 + 0.495i)T + (-5.54 - 52.7i)T^{2} \) |
| 59 | \( 1 + (-0.336 - 0.149i)T + (39.4 + 43.8i)T^{2} \) |
| 61 | \( 1 + (3.35 + 3.72i)T + (-6.37 + 60.6i)T^{2} \) |
| 67 | \( 1 + (-0.451 - 0.781i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (4.59 - 14.1i)T + (-57.4 - 41.7i)T^{2} \) |
| 73 | \( 1 + (7.34 + 3.26i)T + (48.8 + 54.2i)T^{2} \) |
| 79 | \( 1 + (3.96 + 0.843i)T + (72.1 + 32.1i)T^{2} \) |
| 83 | \( 1 + (-1.25 + 3.85i)T + (-67.1 - 48.7i)T^{2} \) |
| 89 | \( 1 + (4.15 - 7.19i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (2.63 + 8.09i)T + (-78.4 + 57.0i)T^{2} \) |
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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
−10.05513688642533636945097697546, −9.009394711154232382242024816951, −8.238855238873304232423411130943, −7.71827925949710747985715019912, −6.87799685718076279705983834299, −5.51554566575060078500767135857, −4.66064055185407468162086708140, −2.82443712343465180402247138796, −1.40322378345834228470092330615, −0.28626312705417749480258771850,
3.13287141603781829600340841047, 3.70239235014028589209490168566, 4.71211635742999029448346395265, 5.59524569263489169971868530443, 7.28866040983338585460222267935, 8.126300507330700900085983244386, 9.022538926367436390968804629835, 9.620104923544816920423331582090, 10.41065896049639030743389733616, 11.14240611830325674071671554133
