| L(s) = 1 | + (0.788 − 1.17i)2-s + (−0.897 − 0.371i)3-s + (−0.757 − 1.85i)4-s + (0.598 − 1.17i)5-s + (−1.14 + 0.760i)6-s + (0.523 − 0.125i)7-s + (−2.77 − 0.569i)8-s + (−1.45 − 1.45i)9-s + (−0.907 − 1.62i)10-s + (1.62 + 1.38i)11-s + (−0.00830 + 1.94i)12-s + (0.521 − 0.319i)13-s + (0.264 − 0.713i)14-s + (−0.973 + 0.831i)15-s + (−2.85 + 2.80i)16-s + (2.03 − 0.160i)17-s + ⋯ |
| L(s) = 1 | + (0.557 − 0.830i)2-s + (−0.517 − 0.214i)3-s + (−0.378 − 0.925i)4-s + (0.267 − 0.525i)5-s + (−0.466 + 0.310i)6-s + (0.197 − 0.0474i)7-s + (−0.979 − 0.201i)8-s + (−0.484 − 0.484i)9-s + (−0.286 − 0.514i)10-s + (0.490 + 0.418i)11-s + (−0.00239 + 0.560i)12-s + (0.144 − 0.0886i)13-s + (0.0707 − 0.190i)14-s + (−0.251 + 0.214i)15-s + (−0.713 + 0.701i)16-s + (0.493 − 0.0388i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 164 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.465 + 0.885i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 164 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.465 + 0.885i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.645479 - 1.06819i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.645479 - 1.06819i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.788 + 1.17i)T \) |
| 41 | \( 1 + (-2.95 - 5.68i)T \) |
| good | 3 | \( 1 + (0.897 + 0.371i)T + (2.12 + 2.12i)T^{2} \) |
| 5 | \( 1 + (-0.598 + 1.17i)T + (-2.93 - 4.04i)T^{2} \) |
| 7 | \( 1 + (-0.523 + 0.125i)T + (6.23 - 3.17i)T^{2} \) |
| 11 | \( 1 + (-1.62 - 1.38i)T + (1.72 + 10.8i)T^{2} \) |
| 13 | \( 1 + (-0.521 + 0.319i)T + (5.90 - 11.5i)T^{2} \) |
| 17 | \( 1 + (-2.03 + 0.160i)T + (16.7 - 2.65i)T^{2} \) |
| 19 | \( 1 + (-1.57 + 2.57i)T + (-8.62 - 16.9i)T^{2} \) |
| 23 | \( 1 + (-3.79 - 2.75i)T + (7.10 + 21.8i)T^{2} \) |
| 29 | \( 1 + (-3.49 - 0.275i)T + (28.6 + 4.53i)T^{2} \) |
| 31 | \( 1 + (1.31 - 4.04i)T + (-25.0 - 18.2i)T^{2} \) |
| 37 | \( 1 + (1.75 + 5.41i)T + (-29.9 + 21.7i)T^{2} \) |
| 43 | \( 1 + (0.859 + 5.42i)T + (-40.8 + 13.2i)T^{2} \) |
| 47 | \( 1 + (2.27 + 0.545i)T + (41.8 + 21.3i)T^{2} \) |
| 53 | \( 1 + (-0.314 + 4.00i)T + (-52.3 - 8.29i)T^{2} \) |
| 59 | \( 1 + (0.382 - 0.526i)T + (-18.2 - 56.1i)T^{2} \) |
| 61 | \( 1 + (-0.0679 + 0.429i)T + (-58.0 - 18.8i)T^{2} \) |
| 67 | \( 1 + (7.38 + 8.64i)T + (-10.4 + 66.1i)T^{2} \) |
| 71 | \( 1 + (7.28 - 8.52i)T + (-11.1 - 70.1i)T^{2} \) |
| 73 | \( 1 + (11.7 - 11.7i)T - 73iT^{2} \) |
| 79 | \( 1 + (-5.93 + 14.3i)T + (-55.8 - 55.8i)T^{2} \) |
| 83 | \( 1 - 3.22iT - 83T^{2} \) |
| 89 | \( 1 + (-3.46 - 14.4i)T + (-79.2 + 40.4i)T^{2} \) |
| 97 | \( 1 + (3.18 + 3.73i)T + (-15.1 + 95.8i)T^{2} \) |
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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
−12.42517814805664509127920237300, −11.66101260257693248708549009920, −10.83363692688320437702895628506, −9.562111196269112025532951700311, −8.801767455787471994357656111905, −6.93998250146129714749320041593, −5.70632739389672472117089811994, −4.81726699539003090817128450347, −3.24036459745057359336954086415, −1.25100589708836220216747206792,
3.02197059503885573427750818079, 4.59065651340017651020851002711, 5.73717171731526620825500149967, 6.54560725249901889399624271148, 7.84458970793407180579099311170, 8.852742016863512930169072287791, 10.26754615609332724850037995721, 11.35173930202199428611607150420, 12.17682287658679940426090021995, 13.42036816884221006312593570305
