| L(s) = 1 | + (−1.14 − 0.835i)2-s + (0.309 + 0.951i)3-s + (0.00606 + 0.0186i)4-s + (−1.87 − 1.21i)5-s + (0.439 − 1.35i)6-s + 7-s + (−0.869 + 2.67i)8-s + (−0.809 + 0.587i)9-s + (1.13 + 2.96i)10-s + (3.85 + 2.79i)11-s + (−0.0158 + 0.0115i)12-s + (1.83 − 1.33i)13-s + (−1.14 − 0.835i)14-s + (0.579 − 2.15i)15-s + (3.26 − 2.37i)16-s + (0.0918 − 0.282i)17-s + ⋯ |
| L(s) = 1 | + (−0.812 − 0.590i)2-s + (0.178 + 0.549i)3-s + (0.00303 + 0.00932i)4-s + (−0.838 − 0.544i)5-s + (0.179 − 0.551i)6-s + 0.377·7-s + (−0.307 + 0.946i)8-s + (−0.269 + 0.195i)9-s + (0.359 + 0.938i)10-s + (1.16 + 0.844i)11-s + (−0.00458 + 0.00332i)12-s + (0.507 − 0.368i)13-s + (−0.307 − 0.223i)14-s + (0.149 − 0.557i)15-s + (0.816 − 0.593i)16-s + (0.0222 − 0.0685i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.288 + 0.957i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.288 + 0.957i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.682661 - 0.507153i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.682661 - 0.507153i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-0.309 - 0.951i)T \) |
| 5 | \( 1 + (1.87 + 1.21i)T \) |
| 7 | \( 1 - T \) |
| good | 2 | \( 1 + (1.14 + 0.835i)T + (0.618 + 1.90i)T^{2} \) |
| 11 | \( 1 + (-3.85 - 2.79i)T + (3.39 + 10.4i)T^{2} \) |
| 13 | \( 1 + (-1.83 + 1.33i)T + (4.01 - 12.3i)T^{2} \) |
| 17 | \( 1 + (-0.0918 + 0.282i)T + (-13.7 - 9.99i)T^{2} \) |
| 19 | \( 1 + (-1.81 + 5.58i)T + (-15.3 - 11.1i)T^{2} \) |
| 23 | \( 1 + (5.42 + 3.94i)T + (7.10 + 21.8i)T^{2} \) |
| 29 | \( 1 + (0.542 + 1.66i)T + (-23.4 + 17.0i)T^{2} \) |
| 31 | \( 1 + (-2.23 + 6.89i)T + (-25.0 - 18.2i)T^{2} \) |
| 37 | \( 1 + (-4.95 + 3.59i)T + (11.4 - 35.1i)T^{2} \) |
| 41 | \( 1 + (-7.97 + 5.79i)T + (12.6 - 38.9i)T^{2} \) |
| 43 | \( 1 + 0.122T + 43T^{2} \) |
| 47 | \( 1 + (-1.80 - 5.56i)T + (-38.0 + 27.6i)T^{2} \) |
| 53 | \( 1 + (1.60 + 4.94i)T + (-42.8 + 31.1i)T^{2} \) |
| 59 | \( 1 + (-1.24 + 0.903i)T + (18.2 - 56.1i)T^{2} \) |
| 61 | \( 1 + (-3.89 - 2.83i)T + (18.8 + 58.0i)T^{2} \) |
| 67 | \( 1 + (3.36 - 10.3i)T + (-54.2 - 39.3i)T^{2} \) |
| 71 | \( 1 + (0.873 + 2.68i)T + (-57.4 + 41.7i)T^{2} \) |
| 73 | \( 1 + (-7.91 - 5.74i)T + (22.5 + 69.4i)T^{2} \) |
| 79 | \( 1 + (3.83 + 11.8i)T + (-63.9 + 46.4i)T^{2} \) |
| 83 | \( 1 + (-1.24 + 3.82i)T + (-67.1 - 48.7i)T^{2} \) |
| 89 | \( 1 + (-1.57 - 1.14i)T + (27.5 + 84.6i)T^{2} \) |
| 97 | \( 1 + (0.479 + 1.47i)T + (-78.4 + 57.0i)T^{2} \) |
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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
−10.66455157792490765457884420552, −9.629158521733696966620935115853, −9.107292247570491560417249839433, −8.310300155254159897517922623577, −7.45372593233105171449563356051, −5.95761149232510396138885898781, −4.70970869582366248610531847895, −3.97888177710124459337655633365, −2.35557767325848763504688923809, −0.794207569595339592521043019445,
1.23776964565738624653292269000, 3.32712652730583970840088198337, 4.03564126984826775578951963375, 6.02073456108376197127629943555, 6.67573839525161726581002682292, 7.68294633864048373397964078577, 8.180087587752907379859973543770, 8.940847888211824988520691537373, 9.936140822479508162784203389916, 11.12807533641996471713671392428
