L(s) = 1 | + (−1.61 + 0.433i)2-s + (0.700 − 0.404i)4-s + (1.42 − 1.42i)5-s + (0.234 − 2.63i)7-s + (1.41 − 1.41i)8-s + (−1.68 + 2.91i)10-s + (−0.254 − 0.948i)11-s + (−1.60 − 3.22i)13-s + (0.764 + 4.36i)14-s + (−2.48 + 4.29i)16-s + (−2.99 − 5.18i)17-s + (2.71 + 0.726i)19-s + (0.421 − 1.57i)20-s + (0.822 + 1.42i)22-s + (−2.58 − 1.49i)23-s + ⋯ |
L(s) = 1 | + (−1.14 + 0.306i)2-s + (0.350 − 0.202i)4-s + (0.635 − 0.635i)5-s + (0.0885 − 0.996i)7-s + (0.498 − 0.498i)8-s + (−0.532 + 0.922i)10-s + (−0.0766 − 0.285i)11-s + (−0.446 − 0.894i)13-s + (0.204 + 1.16i)14-s + (−0.620 + 1.07i)16-s + (−0.725 − 1.25i)17-s + (0.622 + 0.166i)19-s + (0.0941 − 0.351i)20-s + (0.175 + 0.303i)22-s + (−0.539 − 0.311i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.602 + 0.797i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.602 + 0.797i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.245800 - 0.493902i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.245800 - 0.493902i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 7 | \( 1 + (-0.234 + 2.63i)T \) |
| 13 | \( 1 + (1.60 + 3.22i)T \) |
good | 2 | \( 1 + (1.61 - 0.433i)T + (1.73 - i)T^{2} \) |
| 5 | \( 1 + (-1.42 + 1.42i)T - 5iT^{2} \) |
| 11 | \( 1 + (0.254 + 0.948i)T + (-9.52 + 5.5i)T^{2} \) |
| 17 | \( 1 + (2.99 + 5.18i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (-2.71 - 0.726i)T + (16.4 + 9.5i)T^{2} \) |
| 23 | \( 1 + (2.58 + 1.49i)T + (11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (3.65 - 6.33i)T + (-14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + (6.11 - 6.11i)T - 31iT^{2} \) |
| 37 | \( 1 + (1.24 + 4.63i)T + (-32.0 + 18.5i)T^{2} \) |
| 41 | \( 1 + (-0.886 - 3.30i)T + (-35.5 + 20.5i)T^{2} \) |
| 43 | \( 1 + (0.748 - 0.432i)T + (21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (-2.17 - 2.17i)T + 47iT^{2} \) |
| 53 | \( 1 + 6.32T + 53T^{2} \) |
| 59 | \( 1 + (-0.131 + 0.491i)T + (-51.0 - 29.5i)T^{2} \) |
| 61 | \( 1 + (-11.2 + 6.51i)T + (30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (0.827 - 0.221i)T + (58.0 - 33.5i)T^{2} \) |
| 71 | \( 1 + (-3.01 + 11.2i)T + (-61.4 - 35.5i)T^{2} \) |
| 73 | \( 1 + (1.03 + 1.03i)T + 73iT^{2} \) |
| 79 | \( 1 + 11.6T + 79T^{2} \) |
| 83 | \( 1 + (-1.23 + 1.23i)T - 83iT^{2} \) |
| 89 | \( 1 + (-7.75 + 2.07i)T + (77.0 - 44.5i)T^{2} \) |
| 97 | \( 1 + (12.0 + 3.23i)T + (84.0 + 48.5i)T^{2} \) |
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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
−9.659177605805982643182137412385, −9.218322454748122256215258723178, −8.306487053698351958187700050935, −7.44188739041021430030789203517, −6.90313552149247497790667244572, −5.50574626065318739244486510279, −4.68281094833554174134896921602, −3.36072801210585056580889766985, −1.60565643476544680788938226583, −0.39612196255012959679613141493,
1.89208844160022770839010633227, 2.38274333179179334010579691062, 4.13570357109498190261111585353, 5.40954848891556525653903772548, 6.26262525068356590876948760485, 7.29911163232607058601212079282, 8.229295862785791665124746806022, 9.031098414530109247834316046138, 9.691984953048124195784186597648, 10.23090952573608382532117534841