L(s) = 1 | + (0.866 − 0.5i)2-s + (−0.866 − 0.5i)3-s + (0.499 − 0.866i)4-s + (1.14 + 1.92i)5-s − 0.999·6-s + (4.17 + 2.41i)7-s − 0.999i·8-s + (0.499 + 0.866i)9-s + (1.95 + 1.09i)10-s + 2.00·11-s + (−0.866 + 0.499i)12-s + (−1.65 − 0.957i)13-s + 4.82·14-s + (−0.0310 − 2.23i)15-s + (−0.5 − 0.866i)16-s + (−3.14 + 1.81i)17-s + ⋯ |
L(s) = 1 | + (0.612 − 0.353i)2-s + (−0.499 − 0.288i)3-s + (0.249 − 0.433i)4-s + (0.511 + 0.859i)5-s − 0.408·6-s + (1.57 + 0.911i)7-s − 0.353i·8-s + (0.166 + 0.288i)9-s + (0.617 + 0.345i)10-s + 0.606·11-s + (−0.249 + 0.144i)12-s + (−0.459 − 0.265i)13-s + 1.28·14-s + (−0.00800 − 0.577i)15-s + (−0.125 − 0.216i)16-s + (−0.762 + 0.440i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.945 - 0.326i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.945 - 0.326i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.470046051\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.470046051\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + (-1.14 - 1.92i)T \) |
| 37 | \( 1 + (-0.348 + 6.07i)T \) |
good | 7 | \( 1 + (-4.17 - 2.41i)T + (3.5 + 6.06i)T^{2} \) |
| 11 | \( 1 - 2.00T + 11T^{2} \) |
| 13 | \( 1 + (1.65 + 0.957i)T + (6.5 + 11.2i)T^{2} \) |
| 17 | \( 1 + (3.14 - 1.81i)T + (8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (3.31 - 5.74i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 - 3.73iT - 23T^{2} \) |
| 29 | \( 1 - 9.24T + 29T^{2} \) |
| 31 | \( 1 + 5.11T + 31T^{2} \) |
| 41 | \( 1 + (2.64 - 4.57i)T + (-20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + 10.5iT - 43T^{2} \) |
| 47 | \( 1 + 6.25iT - 47T^{2} \) |
| 53 | \( 1 + (-1.38 + 0.797i)T + (26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (-5.98 - 10.3i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (-4.97 + 8.61i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (6.48 + 3.74i)T + (33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (-6.61 + 11.4i)T + (-35.5 - 61.4i)T^{2} \) |
| 73 | \( 1 + 0.404iT - 73T^{2} \) |
| 79 | \( 1 + (-2.99 + 5.17i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (-5.69 + 3.28i)T + (41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 + (3.53 + 6.12i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 - 3.63iT - 97T^{2} \) |
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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
−10.31888797115610340778974388279, −9.089270325566761399937744080521, −8.195090387113629428522924897816, −7.22611904896251782466838536602, −6.27197266518741593052808024119, −5.62368711857940122169790948747, −4.83962597421070805461644522669, −3.72025922329702187163391440064, −2.23785643265314920579377442329, −1.72658691747329752953820463374,
1.02762452246327772300342323456, 2.35010905643051799295143812953, 4.25915270026540651943946040878, 4.61609195848506126279959623761, 5.17861080100055902619024639826, 6.46298272340360567877076124961, 7.04362593273173778898019401163, 8.232092999382443684331692709967, 8.793428858817030550168340162380, 9.836615357712160033896108320346