L(s) = 1 | + (−2.45 − 0.962i)2-s + (3.61 + 3.35i)4-s + (1.47 + 1.00i)5-s + (2.09 − 1.60i)7-s + (−3.35 − 6.97i)8-s + (−2.65 − 3.89i)10-s + (−0.538 − 3.57i)11-s + (−2.76 + 2.20i)13-s + (−6.69 + 1.92i)14-s + (0.785 + 10.4i)16-s + (2.32 + 0.718i)17-s + (6.52 − 3.76i)19-s + (1.96 + 8.60i)20-s + (−2.11 + 9.28i)22-s + (1.91 + 6.21i)23-s + ⋯ |
L(s) = 1 | + (−1.73 − 0.680i)2-s + (1.80 + 1.67i)4-s + (0.660 + 0.450i)5-s + (0.793 − 0.608i)7-s + (−1.18 − 2.46i)8-s + (−0.839 − 1.23i)10-s + (−0.162 − 1.07i)11-s + (−0.767 + 0.612i)13-s + (−1.79 + 0.514i)14-s + (0.196 + 2.61i)16-s + (0.564 + 0.174i)17-s + (1.49 − 0.864i)19-s + (0.439 + 1.92i)20-s + (−0.451 + 1.97i)22-s + (0.399 + 1.29i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.589 + 0.807i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.589 + 0.807i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.681872 - 0.346579i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.681872 - 0.346579i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 7 | \( 1 + (-2.09 + 1.60i)T \) |
good | 2 | \( 1 + (2.45 + 0.962i)T + (1.46 + 1.36i)T^{2} \) |
| 5 | \( 1 + (-1.47 - 1.00i)T + (1.82 + 4.65i)T^{2} \) |
| 11 | \( 1 + (0.538 + 3.57i)T + (-10.5 + 3.24i)T^{2} \) |
| 13 | \( 1 + (2.76 - 2.20i)T + (2.89 - 12.6i)T^{2} \) |
| 17 | \( 1 + (-2.32 - 0.718i)T + (14.0 + 9.57i)T^{2} \) |
| 19 | \( 1 + (-6.52 + 3.76i)T + (9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (-1.91 - 6.21i)T + (-19.0 + 12.9i)T^{2} \) |
| 29 | \( 1 + (-4.35 + 0.994i)T + (26.1 - 12.5i)T^{2} \) |
| 31 | \( 1 + (5.85 + 3.38i)T + (15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (4.18 - 3.88i)T + (2.76 - 36.8i)T^{2} \) |
| 41 | \( 1 + (-3.53 + 1.70i)T + (25.5 - 32.0i)T^{2} \) |
| 43 | \( 1 + (-3.05 - 1.47i)T + (26.8 + 33.6i)T^{2} \) |
| 47 | \( 1 + (-1.81 + 4.62i)T + (-34.4 - 31.9i)T^{2} \) |
| 53 | \( 1 + (3.20 - 3.45i)T + (-3.96 - 52.8i)T^{2} \) |
| 59 | \( 1 + (-9.48 + 6.46i)T + (21.5 - 54.9i)T^{2} \) |
| 61 | \( 1 + (-4.54 - 4.89i)T + (-4.55 + 60.8i)T^{2} \) |
| 67 | \( 1 + (-6.90 + 11.9i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + (-1.21 - 0.276i)T + (63.9 + 30.8i)T^{2} \) |
| 73 | \( 1 + (-8.19 + 3.21i)T + (53.5 - 49.6i)T^{2} \) |
| 79 | \( 1 + (-5.24 - 9.07i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (8.72 - 10.9i)T + (-18.4 - 80.9i)T^{2} \) |
| 89 | \( 1 + (7.03 + 1.05i)T + (85.0 + 26.2i)T^{2} \) |
| 97 | \( 1 + 1.36iT - 97T^{2} \) |
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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
−10.98078525139123061051936459664, −9.907089464499505032670923721497, −9.468710693341684246498250026868, −8.381236741734963746115186063039, −7.55071994696778969735998620274, −6.85465916065293812124175187080, −5.38373802533072879687071359512, −3.48203754079619802958359475562, −2.32538855539676354314170316185, −1.01955806357107036947473238175,
1.29386798142018723481165912296, 2.42631048724440734956397722575, 5.12726545901711607739738050275, 5.63357653197596803156084320272, 7.04762819225558357258085698646, 7.71016908221751772073097375684, 8.563374063679109061875853956146, 9.442862195520560586826669934679, 9.976687619017131872529208238758, 10.80578973766684947216764795355