L(s) = 1 | + (0.0881 − 0.328i)2-s + (1.63 + 0.942i)4-s + (−0.552 − 2.06i)5-s + (2.31 − 1.28i)7-s + (0.935 − 0.935i)8-s − 0.727·10-s + (0.152 − 0.152i)11-s + (−3.58 + 0.408i)13-s + (−0.219 − 0.873i)14-s + (1.65 + 2.87i)16-s + (2.64 − 4.58i)17-s + (4.54 − 4.54i)19-s + (1.04 − 3.88i)20-s + (−0.0367 − 0.0636i)22-s + (−5.58 + 3.22i)23-s + ⋯ |
L(s) = 1 | + (0.0623 − 0.232i)2-s + (0.815 + 0.471i)4-s + (−0.247 − 0.922i)5-s + (0.873 − 0.486i)7-s + (0.330 − 0.330i)8-s − 0.230·10-s + (0.0460 − 0.0460i)11-s + (−0.993 + 0.113i)13-s + (−0.0586 − 0.233i)14-s + (0.414 + 0.718i)16-s + (0.641 − 1.11i)17-s + (1.04 − 1.04i)19-s + (0.232 − 0.869i)20-s + (−0.00783 − 0.0135i)22-s + (−1.16 + 0.672i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.497 + 0.867i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.497 + 0.867i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.74526 - 1.01103i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.74526 - 1.01103i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 7 | \( 1 + (-2.31 + 1.28i)T \) |
| 13 | \( 1 + (3.58 - 0.408i)T \) |
good | 2 | \( 1 + (-0.0881 + 0.328i)T + (-1.73 - i)T^{2} \) |
| 5 | \( 1 + (0.552 + 2.06i)T + (-4.33 + 2.5i)T^{2} \) |
| 11 | \( 1 + (-0.152 + 0.152i)T - 11iT^{2} \) |
| 17 | \( 1 + (-2.64 + 4.58i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (-4.54 + 4.54i)T - 19iT^{2} \) |
| 23 | \( 1 + (5.58 - 3.22i)T + (11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (1.21 - 2.10i)T + (-14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + (2.54 + 0.681i)T + (26.8 + 15.5i)T^{2} \) |
| 37 | \( 1 + (3.73 + 1.00i)T + (32.0 + 18.5i)T^{2} \) |
| 41 | \( 1 + (-2.28 - 8.53i)T + (-35.5 + 20.5i)T^{2} \) |
| 43 | \( 1 + (4.95 - 2.85i)T + (21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (-12.3 + 3.31i)T + (40.7 - 23.5i)T^{2} \) |
| 53 | \( 1 + (1.97 + 3.42i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (-11.8 + 3.16i)T + (51.0 - 29.5i)T^{2} \) |
| 61 | \( 1 + 2.81iT - 61T^{2} \) |
| 67 | \( 1 + (-1.19 - 1.19i)T + 67iT^{2} \) |
| 71 | \( 1 + (0.653 - 2.44i)T + (-61.4 - 35.5i)T^{2} \) |
| 73 | \( 1 + (-0.00884 + 0.0329i)T + (-63.2 - 36.5i)T^{2} \) |
| 79 | \( 1 + (-7.28 + 12.6i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (7.72 - 7.72i)T - 83iT^{2} \) |
| 89 | \( 1 + (2.16 - 8.08i)T + (-77.0 - 44.5i)T^{2} \) |
| 97 | \( 1 + (7.90 + 2.11i)T + (84.0 + 48.5i)T^{2} \) |
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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
−10.10271383634305154219529919801, −9.307096247016275178549418341178, −8.208033705998716598784274826649, −7.52730611093050169440636598620, −6.96634184323481218915460670255, −5.39906190741791772509901579585, −4.73180700969194627613388725614, −3.61526538104799485361127463568, −2.36949776188507655817377392633, −1.05345412683154665228663798608,
1.70405702830551151408916130482, 2.67154106402759313504726982145, 3.95121323744893520963127516725, 5.39694749847102829259302357814, 5.88339547277481069994444823213, 7.07425284778860273930682294988, 7.61600407485164084607030076382, 8.428477710911736095967932105745, 9.835593766331440935761624759007, 10.44314620816606559628298910055