L(s) = 1 | − i·2-s − 3-s − 4-s + (0.866 − 0.5i)5-s + i·6-s + (−0.866 + 0.5i)7-s + i·8-s + 9-s + (−0.5 − 0.866i)10-s + (0.5 + 0.866i)11-s + 12-s + (0.5 + 0.866i)14-s + (−0.866 + 0.5i)15-s + 16-s + (−0.5 + 0.866i)17-s − i·18-s + ⋯ |
L(s) = 1 | − i·2-s − 3-s − 4-s + (0.866 − 0.5i)5-s + i·6-s + (−0.866 + 0.5i)7-s + i·8-s + 9-s + (−0.5 − 0.866i)10-s + (0.5 + 0.866i)11-s + 12-s + (0.5 + 0.866i)14-s + (−0.866 + 0.5i)15-s + 16-s + (−0.5 + 0.866i)17-s − i·18-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1368 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.790 + 0.612i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1368 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.790 + 0.612i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.7663312081\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.7663312081\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + iT \) |
| 3 | \( 1 + T \) |
| 19 | \( 1 - T \) |
good | 5 | \( 1 + (-0.866 + 0.5i)T + (0.5 - 0.866i)T^{2} \) |
| 7 | \( 1 + (0.866 - 0.5i)T + (0.5 - 0.866i)T^{2} \) |
| 11 | \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \) |
| 13 | \( 1 - T^{2} \) |
| 17 | \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \) |
| 23 | \( 1 - T^{2} \) |
| 29 | \( 1 + (-0.866 - 0.5i)T + (0.5 + 0.866i)T^{2} \) |
| 31 | \( 1 + (-0.866 - 0.5i)T + (0.5 + 0.866i)T^{2} \) |
| 37 | \( 1 + 2iT - T^{2} \) |
| 41 | \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \) |
| 43 | \( 1 + T^{2} \) |
| 47 | \( 1 + (0.866 + 0.5i)T + (0.5 + 0.866i)T^{2} \) |
| 53 | \( 1 + (-0.866 + 0.5i)T + (0.5 - 0.866i)T^{2} \) |
| 59 | \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \) |
| 61 | \( 1 + (-0.866 - 0.5i)T + (0.5 + 0.866i)T^{2} \) |
| 67 | \( 1 - 2T + T^{2} \) |
| 71 | \( 1 + (0.866 + 0.5i)T + (0.5 + 0.866i)T^{2} \) |
| 73 | \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \) |
| 79 | \( 1 - T^{2} \) |
| 83 | \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \) |
| 89 | \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \) |
| 97 | \( 1 + T^{2} \) |
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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
−9.706524933102207213809161185769, −9.406618111709471746522543570308, −8.420597764420187083454370283684, −7.05817390044080873027146028106, −6.16295610399732562447420121596, −5.43551125392601401274057227010, −4.66874248408021146650973203897, −3.65820971692902584889112492813, −2.28048000810519457532227374482, −1.25703114398186467022443669450,
0.891104969671182459887282015517, 2.99455399907437500758744137668, 4.15116533850862566982755605435, 5.12940009761133881595293212754, 6.00881636040310916099743565699, 6.51697632667045725581370962330, 7.00975701338042060212418530383, 8.076604005165994214533791791412, 9.238362244819741113297969824314, 9.888207920314216324514564461299