| L(s) = 1 | + (0.156 + 1.40i)2-s + (−0.457 + 1.10i)3-s + (−1.95 + 0.439i)4-s + (−0.453 + 0.891i)5-s + (−1.62 − 0.470i)6-s + (0.196 + 0.819i)7-s + (−0.922 − 2.67i)8-s + (1.10 + 1.10i)9-s + (−1.32 − 0.498i)10-s + (−3.63 + 4.25i)11-s + (0.407 − 2.35i)12-s + (1.58 + 2.58i)13-s + (−1.12 + 0.404i)14-s + (−0.776 − 0.909i)15-s + (3.61 − 1.71i)16-s + (−0.0556 − 0.706i)17-s + ⋯ |
| L(s) = 1 | + (0.110 + 0.993i)2-s + (−0.264 + 0.638i)3-s + (−0.975 + 0.219i)4-s + (−0.203 + 0.398i)5-s + (−0.663 − 0.192i)6-s + (0.0743 + 0.309i)7-s + (−0.326 − 0.945i)8-s + (0.369 + 0.369i)9-s + (−0.418 − 0.157i)10-s + (−1.09 + 1.28i)11-s + (0.117 − 0.680i)12-s + (0.438 + 0.716i)13-s + (−0.299 + 0.108i)14-s + (−0.200 − 0.234i)15-s + (0.903 − 0.428i)16-s + (−0.0134 − 0.171i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 820 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.457 + 0.889i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 820 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.457 + 0.889i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.416032 - 0.681804i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.416032 - 0.681804i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.156 - 1.40i)T \) |
| 5 | \( 1 + (0.453 - 0.891i)T \) |
| 41 | \( 1 + (6.17 + 1.68i)T \) |
| good | 3 | \( 1 + (0.457 - 1.10i)T + (-2.12 - 2.12i)T^{2} \) |
| 7 | \( 1 + (-0.196 - 0.819i)T + (-6.23 + 3.17i)T^{2} \) |
| 11 | \( 1 + (3.63 - 4.25i)T + (-1.72 - 10.8i)T^{2} \) |
| 13 | \( 1 + (-1.58 - 2.58i)T + (-5.90 + 11.5i)T^{2} \) |
| 17 | \( 1 + (0.0556 + 0.706i)T + (-16.7 + 2.65i)T^{2} \) |
| 19 | \( 1 + (0.254 + 0.155i)T + (8.62 + 16.9i)T^{2} \) |
| 23 | \( 1 + (3.49 + 2.53i)T + (7.10 + 21.8i)T^{2} \) |
| 29 | \( 1 + (-0.332 + 4.22i)T + (-28.6 - 4.53i)T^{2} \) |
| 31 | \( 1 + (-1.49 + 4.58i)T + (-25.0 - 18.2i)T^{2} \) |
| 37 | \( 1 + (2.25 + 6.92i)T + (-29.9 + 21.7i)T^{2} \) |
| 43 | \( 1 + (-1.68 - 10.6i)T + (-40.8 + 13.2i)T^{2} \) |
| 47 | \( 1 + (1.55 - 6.47i)T + (-41.8 - 21.3i)T^{2} \) |
| 53 | \( 1 + (-6.21 - 0.489i)T + (52.3 + 8.29i)T^{2} \) |
| 59 | \( 1 + (-4.46 + 6.15i)T + (-18.2 - 56.1i)T^{2} \) |
| 61 | \( 1 + (1.69 - 10.6i)T + (-58.0 - 18.8i)T^{2} \) |
| 67 | \( 1 + (-0.307 + 0.263i)T + (10.4 - 66.1i)T^{2} \) |
| 71 | \( 1 + (9.41 + 8.03i)T + (11.1 + 70.1i)T^{2} \) |
| 73 | \( 1 + (-4.79 + 4.79i)T - 73iT^{2} \) |
| 79 | \( 1 + (1.09 + 0.453i)T + (55.8 + 55.8i)T^{2} \) |
| 83 | \( 1 - 10.8iT - 83T^{2} \) |
| 89 | \( 1 + (2.86 - 0.688i)T + (79.2 - 40.4i)T^{2} \) |
| 97 | \( 1 + (11.5 - 9.84i)T + (15.1 - 95.8i)T^{2} \) |
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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
−10.49148562158608209453420832401, −9.937279806514430415195841122728, −9.135761946420155626983406044497, −7.987605836458994698036617343022, −7.43995906160629818901488988807, −6.47302836293996189053705895271, −5.48591776632459685377327991712, −4.60177708029901182173309192383, −3.99626722613530892750080105974, −2.32027908750738039343988019580,
0.40807277694756703764935490321, 1.51752375254472954186256071554, 3.06413037559070864048310462866, 3.89966849307511299650852256901, 5.21800459446944401588790468917, 5.85934940747194075120548924253, 7.15531002509040897175685157214, 8.285098136560408922286484069853, 8.638181435995834668133592231422, 10.02405883930967477376608416009
