| L(s) = 1 | + (0.939 − 0.342i)2-s + (0.00627 − 0.0355i)3-s + (0.766 − 0.642i)4-s + (0.766 + 0.642i)5-s + (−0.00627 − 0.0355i)6-s + (−0.918 − 1.59i)7-s + (0.500 − 0.866i)8-s + (2.81 + 1.02i)9-s + (0.939 + 0.342i)10-s + (1.23 − 2.13i)11-s + (−0.0180 − 0.0313i)12-s + (0.415 + 2.35i)13-s + (−1.40 − 1.18i)14-s + (0.0276 − 0.0232i)15-s + (0.173 − 0.984i)16-s + (−6.33 + 2.30i)17-s + ⋯ |
| L(s) = 1 | + (0.664 − 0.241i)2-s + (0.00362 − 0.0205i)3-s + (0.383 − 0.321i)4-s + (0.342 + 0.287i)5-s + (−0.00256 − 0.0145i)6-s + (−0.347 − 0.601i)7-s + (0.176 − 0.306i)8-s + (0.939 + 0.341i)9-s + (0.297 + 0.108i)10-s + (0.371 − 0.643i)11-s + (−0.00521 − 0.00903i)12-s + (0.115 + 0.653i)13-s + (−0.376 − 0.315i)14-s + (0.00714 − 0.00599i)15-s + (0.0434 − 0.246i)16-s + (−1.53 + 0.559i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 190 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.914 + 0.404i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 190 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.914 + 0.404i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.70967 - 0.360943i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.70967 - 0.360943i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.939 + 0.342i)T \) |
| 5 | \( 1 + (-0.766 - 0.642i)T \) |
| 19 | \( 1 + (4.34 - 0.298i)T \) |
| good | 3 | \( 1 + (-0.00627 + 0.0355i)T + (-2.81 - 1.02i)T^{2} \) |
| 7 | \( 1 + (0.918 + 1.59i)T + (-3.5 + 6.06i)T^{2} \) |
| 11 | \( 1 + (-1.23 + 2.13i)T + (-5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (-0.415 - 2.35i)T + (-12.2 + 4.44i)T^{2} \) |
| 17 | \( 1 + (6.33 - 2.30i)T + (13.0 - 10.9i)T^{2} \) |
| 23 | \( 1 + (-1.24 + 1.04i)T + (3.99 - 22.6i)T^{2} \) |
| 29 | \( 1 + (3.10 + 1.12i)T + (22.2 + 18.6i)T^{2} \) |
| 31 | \( 1 + (-1.75 - 3.03i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + 6.00T + 37T^{2} \) |
| 41 | \( 1 + (1.38 - 7.85i)T + (-38.5 - 14.0i)T^{2} \) |
| 43 | \( 1 + (-4.37 - 3.67i)T + (7.46 + 42.3i)T^{2} \) |
| 47 | \( 1 + (11.7 + 4.27i)T + (36.0 + 30.2i)T^{2} \) |
| 53 | \( 1 + (1.47 - 1.23i)T + (9.20 - 52.1i)T^{2} \) |
| 59 | \( 1 + (-4.46 + 1.62i)T + (45.1 - 37.9i)T^{2} \) |
| 61 | \( 1 + (-10.4 + 8.78i)T + (10.5 - 60.0i)T^{2} \) |
| 67 | \( 1 + (-3.49 - 1.27i)T + (51.3 + 43.0i)T^{2} \) |
| 71 | \( 1 + (-4.46 - 3.74i)T + (12.3 + 69.9i)T^{2} \) |
| 73 | \( 1 + (0.0886 - 0.502i)T + (-68.5 - 24.9i)T^{2} \) |
| 79 | \( 1 + (-2.10 + 11.9i)T + (-74.2 - 27.0i)T^{2} \) |
| 83 | \( 1 + (3.11 + 5.39i)T + (-41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 + (-2.95 - 16.7i)T + (-83.6 + 30.4i)T^{2} \) |
| 97 | \( 1 + (-12.8 + 4.67i)T + (74.3 - 62.3i)T^{2} \) |
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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
−12.85679045592544362968700159995, −11.40345405643523160900147823798, −10.70043612954315243659122220819, −9.779745979845375529601880419742, −8.500512646740704049052205288010, −6.88256089656427363948076783629, −6.39059633144509163395666522404, −4.69635543657869100849956282847, −3.70952112232813029665453131201, −1.93479151852949990431148058137,
2.21554340582056243304813703371, 3.96462896839289566046761946984, 5.07507508346999410522835872311, 6.35164136597796450089010182407, 7.16752264243759248936771552918, 8.682364468009086539369136525306, 9.570352084890643088411577697125, 10.73317496289173537960528121243, 11.95465359855730984569660252024, 12.84774534041614501833266443616
