L(s) = 1 | + (−0.410 + 1.35i)2-s + (2.26 − 1.30i)3-s + (−1.66 − 1.11i)4-s + (−0.5 + 0.866i)5-s + (0.841 + 3.60i)6-s + (1.72 + 2.00i)7-s + (2.18 − 1.79i)8-s + (1.92 − 3.33i)9-s + (−0.967 − 1.03i)10-s + (0.530 + 0.919i)11-s + (−5.22 − 0.338i)12-s + 0.831·13-s + (−3.42 + 1.51i)14-s + 2.61i·15-s + (1.53 + 3.69i)16-s + (4.14 − 2.39i)17-s + ⋯ |
L(s) = 1 | + (−0.289 + 0.957i)2-s + (1.30 − 0.755i)3-s + (−0.831 − 0.555i)4-s + (−0.223 + 0.387i)5-s + (0.343 + 1.47i)6-s + (0.652 + 0.757i)7-s + (0.772 − 0.635i)8-s + (0.642 − 1.11i)9-s + (−0.305 − 0.326i)10-s + (0.160 + 0.277i)11-s + (−1.50 − 0.0978i)12-s + 0.230·13-s + (−0.914 + 0.404i)14-s + 0.675i·15-s + (0.383 + 0.923i)16-s + (1.00 − 0.580i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.732 - 0.680i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 280 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.732 - 0.680i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.48008 + 0.581456i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.48008 + 0.581456i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.410 - 1.35i)T \) |
| 5 | \( 1 + (0.5 - 0.866i)T \) |
| 7 | \( 1 + (-1.72 - 2.00i)T \) |
good | 3 | \( 1 + (-2.26 + 1.30i)T + (1.5 - 2.59i)T^{2} \) |
| 11 | \( 1 + (-0.530 - 0.919i)T + (-5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 - 0.831T + 13T^{2} \) |
| 17 | \( 1 + (-4.14 + 2.39i)T + (8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (2.03 + 1.17i)T + (9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (-1.32 - 0.763i)T + (11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 - 3.07iT - 29T^{2} \) |
| 31 | \( 1 + (4.78 + 8.28i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (10.0 + 5.81i)T + (18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 - 8.76iT - 41T^{2} \) |
| 43 | \( 1 + 4.99T + 43T^{2} \) |
| 47 | \( 1 + (1.75 - 3.04i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (6.61 - 3.81i)T + (26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (3.31 - 1.91i)T + (29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (-6.51 + 11.2i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (7.36 + 12.7i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + 9.99iT - 71T^{2} \) |
| 73 | \( 1 + (7.06 - 4.07i)T + (36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (-8.17 - 4.72i)T + (39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + 13.7iT - 83T^{2} \) |
| 89 | \( 1 + (-10.1 - 5.88i)T + (44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 - 6.01iT - 97T^{2} \) |
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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
−12.24048319051936314957822206717, −10.93932393348827626331312302994, −9.522322751739552417767922342189, −8.883368820565383258108270250275, −7.924883882769525299409861152169, −7.45088820286686706448054297775, −6.31385161635354467861932449023, −4.99153377185805417477704202882, −3.38374415335117046904347653669, −1.81320819915492317385829758045,
1.62955813668419869500165004546, 3.32664232569494199255076503504, 3.97980271068932704733996828691, 5.08774349760418832702128751471, 7.39757787616750255231954858053, 8.511490085475371681948154864900, 8.674758041005754090912895756263, 10.05707910602101385896347334791, 10.46846210998527069501801677697, 11.62070881205760634127724418799