L(s) = 1 | + (0.866 + 0.5i)2-s + (0.866 + 0.5i)3-s + (0.499 + 0.866i)4-s + (0.499 + 0.866i)6-s + (−1.00 + 0.581i)7-s + 0.999i·8-s + (0.499 + 0.866i)9-s + (−0.5 + 0.866i)11-s + 0.999i·12-s + (−3.60 + 0.0811i)13-s − 1.16·14-s + (−0.5 + 0.866i)16-s + (−1.00 + 0.581i)17-s + 0.999i·18-s + (2.58 + 4.47i)19-s + ⋯ |
L(s) = 1 | + (0.612 + 0.353i)2-s + (0.499 + 0.288i)3-s + (0.249 + 0.433i)4-s + (0.204 + 0.353i)6-s + (−0.380 + 0.219i)7-s + 0.353i·8-s + (0.166 + 0.288i)9-s + (−0.150 + 0.261i)11-s + 0.288i·12-s + (−0.999 + 0.0225i)13-s − 0.310·14-s + (−0.125 + 0.216i)16-s + (−0.244 + 0.140i)17-s + 0.235i·18-s + (0.592 + 1.02i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.758 - 0.651i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.758 - 0.651i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.170160180\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.170160180\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + (-0.866 - 0.5i)T \) |
| 5 | \( 1 \) |
| 13 | \( 1 + (3.60 - 0.0811i)T \) |
good | 7 | \( 1 + (1.00 - 0.581i)T + (3.5 - 6.06i)T^{2} \) |
| 11 | \( 1 + (0.5 - 0.866i)T + (-5.5 - 9.52i)T^{2} \) |
| 17 | \( 1 + (1.00 - 0.581i)T + (8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (-2.58 - 4.47i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (-1.87 - 1.08i)T + (11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (4.16 - 7.20i)T + (-14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + 0.837T + 31T^{2} \) |
| 37 | \( 1 + (-0.140 - 0.0811i)T + (18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (-0.581 + 1.00i)T + (-20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (2.73 - 1.58i)T + (21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 - 6iT - 47T^{2} \) |
| 53 | \( 1 + 7.48iT - 53T^{2} \) |
| 59 | \( 1 + (-5 - 8.66i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (-0.0811 - 0.140i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (5.24 + 9.08i)T + (-35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + 3.32iT - 73T^{2} \) |
| 79 | \( 1 - 13.4T + 79T^{2} \) |
| 83 | \( 1 + 9iT - 83T^{2} \) |
| 89 | \( 1 + (5.74 - 9.94i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (-4.61 + 2.66i)T + (48.5 - 84.0i)T^{2} \) |
show more | |
show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.434733896469560250959905734672, −8.716697107222669859913657598543, −7.70795516916573436288756341217, −7.25909187436178690880988327071, −6.26608737351583674148716115578, −5.35609252635692625209791514701, −4.68680536329369321301016148620, −3.63936958021492730524361375262, −2.93093649607611574082218205114, −1.82081042733772559398909826865,
0.55565844508511443690418902850, 2.11883696465707211006967032864, 2.87120781307994414540673102261, 3.77711848284127417483017648515, 4.74743341096603098167044255245, 5.53769264754410240146208639612, 6.61457857342215077156392635069, 7.19452303358810176810673421561, 8.005510651948186263045950203287, 9.041813172279831436028227225183