L(s) = 1 | + (0.5 − 0.866i)2-s + (0.5 − 0.866i)3-s + (−0.499 − 0.866i)4-s + 4·5-s + (−0.499 − 0.866i)6-s + (−0.5 − 0.866i)7-s − 0.999·8-s + (−0.499 − 0.866i)9-s + (2 − 3.46i)10-s + (1.5 − 2.59i)11-s − 0.999·12-s + (−2.5 + 2.59i)13-s − 0.999·14-s + (2 − 3.46i)15-s + (−0.5 + 0.866i)16-s + (2.5 + 4.33i)17-s + ⋯ |
L(s) = 1 | + (0.353 − 0.612i)2-s + (0.288 − 0.499i)3-s + (−0.249 − 0.433i)4-s + 1.78·5-s + (−0.204 − 0.353i)6-s + (−0.188 − 0.327i)7-s − 0.353·8-s + (−0.166 − 0.288i)9-s + (0.632 − 1.09i)10-s + (0.452 − 0.783i)11-s − 0.288·12-s + (−0.693 + 0.720i)13-s − 0.267·14-s + (0.516 − 0.894i)15-s + (−0.125 + 0.216i)16-s + (0.606 + 1.05i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0128 + 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0128 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.67330 - 1.65198i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.67330 - 1.65198i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.5 + 0.866i)T \) |
| 3 | \( 1 + (-0.5 + 0.866i)T \) |
| 7 | \( 1 + (0.5 + 0.866i)T \) |
| 13 | \( 1 + (2.5 - 2.59i)T \) |
good | 5 | \( 1 - 4T + 5T^{2} \) |
| 11 | \( 1 + (-1.5 + 2.59i)T + (-5.5 - 9.52i)T^{2} \) |
| 17 | \( 1 + (-2.5 - 4.33i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (1.5 + 2.59i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (3 - 5.19i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (-4.5 + 7.79i)T + (-14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + 4T + 31T^{2} \) |
| 37 | \( 1 + (2 - 3.46i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (2.5 - 4.33i)T + (-20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (-21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + 3T + 47T^{2} \) |
| 53 | \( 1 + 11T + 53T^{2} \) |
| 59 | \( 1 + (1 + 1.73i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (-0.5 - 0.866i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (1 - 1.73i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + (3 + 5.19i)T + (-35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 - 12T + 73T^{2} \) |
| 79 | \( 1 - 11T + 79T^{2} \) |
| 83 | \( 1 - 6T + 83T^{2} \) |
| 89 | \( 1 + (3.5 - 6.06i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (-6 - 10.3i)T + (-48.5 + 84.0i)T^{2} \) |
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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
−10.48036719555506605811388077778, −9.702530217671609123200659693485, −9.182987224855806064294159330136, −8.013122467384839701530357397457, −6.51905856959987526073644511782, −6.12047470895864318007447049586, −4.98948553865467236865492509446, −3.56419567736335344125202481975, −2.32812910948998003646911401521, −1.41657840200847134643381722163,
2.06651423055000875473771703220, 3.15942315338852056502104693645, 4.77580161198113373617957070828, 5.40245095979644550487900414957, 6.32663603249249366271002008719, 7.23088041632629343234814940258, 8.503923422612367660369418439499, 9.387213486985284318908274632670, 9.909368445830808424276765756360, 10.65234761240300713044289414016