L(s) = 1 | + (0.228 − 0.703i)2-s − 2.48·3-s + (1.17 + 0.854i)4-s + (−2.55 − 1.85i)5-s + (−0.568 + 1.75i)6-s + (0.309 + 0.951i)7-s + (2.06 − 1.50i)8-s + 3.19·9-s + (−1.88 + 1.37i)10-s + (−3.23 + 2.35i)11-s + (−2.92 − 2.12i)12-s + (−1.58 + 4.88i)13-s + 0.739·14-s + (6.36 + 4.62i)15-s + (0.314 + 0.969i)16-s + (−4.58 + 3.33i)17-s + ⋯ |
L(s) = 1 | + (0.161 − 0.497i)2-s − 1.43·3-s + (0.587 + 0.427i)4-s + (−1.14 − 0.830i)5-s + (−0.232 + 0.714i)6-s + (0.116 + 0.359i)7-s + (0.730 − 0.530i)8-s + 1.06·9-s + (−0.597 + 0.434i)10-s + (−0.975 + 0.709i)11-s + (−0.844 − 0.613i)12-s + (−0.440 + 1.35i)13-s + 0.197·14-s + (1.64 + 1.19i)15-s + (0.0787 + 0.242i)16-s + (−1.11 + 0.807i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 287 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.325 - 0.945i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 287 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.325 - 0.945i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.185414 + 0.260027i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.185414 + 0.260027i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 7 | \( 1 + (-0.309 - 0.951i)T \) |
| 41 | \( 1 + (-4.70 + 4.33i)T \) |
good | 2 | \( 1 + (-0.228 + 0.703i)T + (-1.61 - 1.17i)T^{2} \) |
| 3 | \( 1 + 2.48T + 3T^{2} \) |
| 5 | \( 1 + (2.55 + 1.85i)T + (1.54 + 4.75i)T^{2} \) |
| 11 | \( 1 + (3.23 - 2.35i)T + (3.39 - 10.4i)T^{2} \) |
| 13 | \( 1 + (1.58 - 4.88i)T + (-10.5 - 7.64i)T^{2} \) |
| 17 | \( 1 + (4.58 - 3.33i)T + (5.25 - 16.1i)T^{2} \) |
| 19 | \( 1 + (0.188 + 0.579i)T + (-15.3 + 11.1i)T^{2} \) |
| 23 | \( 1 + (1.65 - 5.08i)T + (-18.6 - 13.5i)T^{2} \) |
| 29 | \( 1 + (7.91 + 5.75i)T + (8.96 + 27.5i)T^{2} \) |
| 31 | \( 1 + (-5.18 + 3.76i)T + (9.57 - 29.4i)T^{2} \) |
| 37 | \( 1 + (2.83 + 2.06i)T + (11.4 + 35.1i)T^{2} \) |
| 43 | \( 1 + (1.16 - 3.59i)T + (-34.7 - 25.2i)T^{2} \) |
| 47 | \( 1 + (2.32 - 7.16i)T + (-38.0 - 27.6i)T^{2} \) |
| 53 | \( 1 + (6.29 + 4.57i)T + (16.3 + 50.4i)T^{2} \) |
| 59 | \( 1 + (-0.413 + 1.27i)T + (-47.7 - 34.6i)T^{2} \) |
| 61 | \( 1 + (-2.33 - 7.17i)T + (-49.3 + 35.8i)T^{2} \) |
| 67 | \( 1 + (7.11 + 5.17i)T + (20.7 + 63.7i)T^{2} \) |
| 71 | \( 1 + (-3.25 + 2.36i)T + (21.9 - 67.5i)T^{2} \) |
| 73 | \( 1 + 13.6T + 73T^{2} \) |
| 79 | \( 1 - 4.59T + 79T^{2} \) |
| 83 | \( 1 - 11.0T + 83T^{2} \) |
| 89 | \( 1 + (-2.23 - 6.87i)T + (-72.0 + 52.3i)T^{2} \) |
| 97 | \( 1 + (5.42 + 3.94i)T + (29.9 + 92.2i)T^{2} \) |
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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
−11.85571994438407524935482243107, −11.49043204591634049359148944159, −10.74071800558344332528799527284, −9.454811619957480879021059218323, −8.031827303585930543615467560675, −7.24161080918458615327766723326, −6.08933009996092779221711012170, −4.73030583953038808865784478149, −4.11471632282402386077304655431, −2.00758395280424046449160044631,
0.25254390927774214905683516314, 2.94589413716086108057186206061, 4.73435769486745691922435289302, 5.55271241913749946505033877569, 6.61242980697807199223327137263, 7.31111668638312933620819158293, 8.178771044782946371079631065345, 10.35681784325480130044084015121, 10.75654809613005740129164752112, 11.29783697281670161947044616491