| L(s) = 1 | + (−0.601 + 2.04i)2-s + (−0.160 + 1.72i)3-s + (−2.14 − 1.38i)4-s + (−0.358 + 2.49i)5-s + (−3.43 − 1.36i)6-s + (−2.54 + 0.719i)7-s + (0.896 − 0.776i)8-s + (−2.94 − 0.552i)9-s + (−4.88 − 2.23i)10-s + (0.357 + 1.21i)11-s + (2.72 − 3.48i)12-s + (6.26 + 2.86i)13-s + (0.0574 − 5.64i)14-s + (−4.24 − 1.01i)15-s + (−1.07 − 2.34i)16-s + (4.94 − 3.18i)17-s + ⋯ |
| L(s) = 1 | + (−0.425 + 1.44i)2-s + (−0.0924 + 0.995i)3-s + (−1.07 − 0.690i)4-s + (−0.160 + 1.11i)5-s + (−1.40 − 0.557i)6-s + (−0.962 + 0.271i)7-s + (0.316 − 0.274i)8-s + (−0.982 − 0.184i)9-s + (−1.54 − 0.706i)10-s + (0.107 + 0.366i)11-s + (0.787 − 1.00i)12-s + (1.73 + 0.793i)13-s + (0.0153 − 1.50i)14-s + (−1.09 − 0.262i)15-s + (−0.267 − 0.586i)16-s + (1.20 − 0.771i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0800 + 0.996i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0800 + 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.550514 - 0.508063i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.550514 - 0.508063i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (0.160 - 1.72i)T \) |
| 7 | \( 1 + (2.54 - 0.719i)T \) |
| 23 | \( 1 + (3.20 + 3.57i)T \) |
| good | 2 | \( 1 + (0.601 - 2.04i)T + (-1.68 - 1.08i)T^{2} \) |
| 5 | \( 1 + (0.358 - 2.49i)T + (-4.79 - 1.40i)T^{2} \) |
| 11 | \( 1 + (-0.357 - 1.21i)T + (-9.25 + 5.94i)T^{2} \) |
| 13 | \( 1 + (-6.26 - 2.86i)T + (8.51 + 9.82i)T^{2} \) |
| 17 | \( 1 + (-4.94 + 3.18i)T + (7.06 - 15.4i)T^{2} \) |
| 19 | \( 1 + (2.59 - 4.03i)T + (-7.89 - 17.2i)T^{2} \) |
| 29 | \( 1 + (-2.68 - 4.17i)T + (-12.0 + 26.3i)T^{2} \) |
| 31 | \( 1 + (1.68 - 1.46i)T + (4.41 - 30.6i)T^{2} \) |
| 37 | \( 1 + (0.484 + 3.37i)T + (-35.5 + 10.4i)T^{2} \) |
| 41 | \( 1 + (0.362 - 2.52i)T + (-39.3 - 11.5i)T^{2} \) |
| 43 | \( 1 + (-0.810 + 0.935i)T + (-6.11 - 42.5i)T^{2} \) |
| 47 | \( 1 + 4.02T + 47T^{2} \) |
| 53 | \( 1 + (5.33 - 2.43i)T + (34.7 - 40.0i)T^{2} \) |
| 59 | \( 1 + (1.15 - 2.52i)T + (-38.6 - 44.5i)T^{2} \) |
| 61 | \( 1 + (-2.90 + 2.51i)T + (8.68 - 60.3i)T^{2} \) |
| 67 | \( 1 + (-10.4 - 3.07i)T + (56.3 + 36.2i)T^{2} \) |
| 71 | \( 1 + (-4.05 + 13.8i)T + (-59.7 - 38.3i)T^{2} \) |
| 73 | \( 1 + (8.37 - 13.0i)T + (-30.3 - 66.4i)T^{2} \) |
| 79 | \( 1 + (1.68 - 3.68i)T + (-51.7 - 59.7i)T^{2} \) |
| 83 | \( 1 + (0.734 + 5.10i)T + (-79.6 + 23.3i)T^{2} \) |
| 89 | \( 1 + (8.18 - 9.44i)T + (-12.6 - 88.0i)T^{2} \) |
| 97 | \( 1 + (10.8 + 1.56i)T + (93.0 + 27.3i)T^{2} \) |
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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
−11.39538114726398853757968438002, −10.47062008102648934216900678864, −9.720649286306336649783117932529, −8.901375085689388451212813204024, −8.088577904620492956553109123246, −6.81965529583992685549360807401, −6.32224383309887311125634385626, −5.50028884090722291614366795648, −3.99547395841661776382812241759, −3.02899231875042393883519878830,
0.57212326680893242573649941484, 1.45109033065118518363308535829, 3.05452783817681253912388120132, 3.91053282994385218359053107275, 5.68166759285141247660612057512, 6.47632989165145711588869107392, 8.091742016149297796143615899557, 8.540904598484701760388643164237, 9.483643178774603888903850626001, 10.46781952803526575331894080860
