L(s) = 1 | + (−0.866 − 0.5i)2-s + (0.115 + 1.72i)3-s + (0.499 + 0.866i)4-s + (−1.85 + 3.22i)5-s + (0.763 − 1.55i)6-s + (−1.49 − 2.17i)7-s − 0.999i·8-s + (−2.97 + 0.400i)9-s + (3.22 − 1.85i)10-s + (0.553 − 0.319i)11-s + (−1.43 + 0.964i)12-s + i·13-s + (0.209 + 2.63i)14-s + (−5.78 − 2.84i)15-s + (−0.5 + 0.866i)16-s + (3.63 + 6.30i)17-s + ⋯ |
L(s) = 1 | + (−0.612 − 0.353i)2-s + (0.0668 + 0.997i)3-s + (0.249 + 0.433i)4-s + (−0.831 + 1.44i)5-s + (0.311 − 0.634i)6-s + (−0.566 − 0.823i)7-s − 0.353i·8-s + (−0.991 + 0.133i)9-s + (1.01 − 0.588i)10-s + (0.166 − 0.0963i)11-s + (−0.415 + 0.278i)12-s + 0.277i·13-s + (0.0558 + 0.704i)14-s + (−1.49 − 0.733i)15-s + (−0.125 + 0.216i)16-s + (0.882 + 1.52i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.856 + 0.516i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.856 + 0.516i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.0583289 - 0.209603i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.0583289 - 0.209603i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.866 + 0.5i)T \) |
| 3 | \( 1 + (-0.115 - 1.72i)T \) |
| 7 | \( 1 + (1.49 + 2.17i)T \) |
| 13 | \( 1 - iT \) |
good | 5 | \( 1 + (1.85 - 3.22i)T + (-2.5 - 4.33i)T^{2} \) |
| 11 | \( 1 + (-0.553 + 0.319i)T + (5.5 - 9.52i)T^{2} \) |
| 17 | \( 1 + (-3.63 - 6.30i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (4.62 + 2.67i)T + (9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (4.07 + 2.35i)T + (11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + 2.98iT - 29T^{2} \) |
| 31 | \( 1 + (-0.608 + 0.351i)T + (15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (-5.21 + 9.04i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + 11.6T + 41T^{2} \) |
| 43 | \( 1 + 3.57T + 43T^{2} \) |
| 47 | \( 1 + (-3.53 + 6.12i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (6.65 - 3.84i)T + (26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (-1.35 - 2.34i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (-10.0 - 5.77i)T + (30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (1.15 + 1.99i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 - 10.9iT - 71T^{2} \) |
| 73 | \( 1 + (8.28 - 4.78i)T + (36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (5.00 - 8.67i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + 6.70T + 83T^{2} \) |
| 89 | \( 1 + (1.95 - 3.37i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 - 3.50iT - 97T^{2} \) |
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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
−11.04674715052541437775358810351, −10.23012286815916011809436489735, −10.10516954995123236468935975285, −8.685642473027394734699989936093, −7.912308122731710998783572413680, −6.89288276063621233180199718610, −6.07949276650405003782036272126, −4.04675522451594690613173950769, −3.75722957363745200086704299847, −2.56659605563159690048308660390,
0.15067927709302213717046061886, 1.56995582560048583903038969281, 3.18452064570368490556914400744, 4.88921882190755283786074968145, 5.77461196918116660263307098196, 6.78606475559296099735622965524, 7.893983646378808178116869309401, 8.328940807906066326721376575259, 9.130414056284429287212812737403, 9.943837345460011532130236173814