L(s) = 1 | + (0.866 − 0.5i)2-s + (0.866 + 0.5i)3-s + (0.499 − 0.866i)4-s + (−2.13 − 0.650i)5-s + 0.999·6-s − 0.999i·8-s + (0.499 + 0.866i)9-s + (−2.17 + 0.506i)10-s + (−2.23 + 3.87i)11-s + (0.866 − 0.499i)12-s + 5.88i·13-s + (−1.52 − 1.63i)15-s + (−0.5 − 0.866i)16-s + (−6.69 − 3.86i)17-s + (0.866 + 0.499i)18-s + (3.30 + 5.73i)19-s + ⋯ |
L(s) = 1 | + (0.612 − 0.353i)2-s + (0.499 + 0.288i)3-s + (0.249 − 0.433i)4-s + (−0.956 − 0.290i)5-s + 0.408·6-s − 0.353i·8-s + (0.166 + 0.288i)9-s + (−0.688 + 0.160i)10-s + (−0.673 + 1.16i)11-s + (0.249 − 0.144i)12-s + 1.63i·13-s + (−0.394 − 0.421i)15-s + (−0.125 − 0.216i)16-s + (−1.62 − 0.938i)17-s + (0.204 + 0.117i)18-s + (0.759 + 1.31i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1470 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.220 - 0.975i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1470 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.220 - 0.975i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.699553450\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.699553450\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + (2.13 + 0.650i)T \) |
| 7 | \( 1 \) |
good | 11 | \( 1 + (2.23 - 3.87i)T + (-5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 - 5.88iT - 13T^{2} \) |
| 17 | \( 1 + (6.69 + 3.86i)T + (8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (-3.30 - 5.73i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (-2.26 + 1.30i)T + (11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 - 8.17T + 29T^{2} \) |
| 31 | \( 1 + (4.23 - 7.33i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (2.76 - 1.59i)T + (18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 - 3.56T + 41T^{2} \) |
| 43 | \( 1 - 1.43iT - 43T^{2} \) |
| 47 | \( 1 + (5.88 - 3.39i)T + (23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (-1.19 - 0.690i)T + (26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (-2.33 + 4.03i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (4.89 + 8.48i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (-1.60 - 0.925i)T + (33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 - 2.02T + 71T^{2} \) |
| 73 | \( 1 + (3.47 + 2.00i)T + (36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (3.49 + 6.04i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 - 5.35iT - 83T^{2} \) |
| 89 | \( 1 + (-7.19 - 12.4i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 - 7.71iT - 97T^{2} \) |
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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
−9.602507890144611758990989746501, −9.009344306207209739913638981917, −8.094139634003302539474680769485, −7.14125103578300582133560142613, −6.64012562988585466261412921655, −4.92129724808190579148204638045, −4.68869259989233303562592240712, −3.78975406867242080824409443007, −2.73651002541476139972725977332, −1.66362377236020014000786653145,
0.50312789182697152461837699072, 2.64005154940745889155754090105, 3.17230344404762760706101594482, 4.15184499116948688612281910591, 5.16216347976982468771876570745, 6.08144132983209814694723263694, 6.99813566083998640468614209758, 7.70575628666276615209094462288, 8.384053231209082472874644529855, 8.900037054696774421104858382287