L(s) = 1 | + (0.866 − 0.5i)2-s + (−0.866 − 0.5i)3-s + (0.499 − 0.866i)4-s − 0.999·6-s + (2.59 + 0.5i)7-s − 0.999i·8-s + (0.499 + 0.866i)9-s + (−0.866 + 0.499i)12-s − 4i·13-s + (2.5 − 0.866i)14-s + (−0.5 − 0.866i)16-s + (2.59 + 1.5i)17-s + (0.866 + 0.499i)18-s + (1 + 1.73i)19-s + (−2 − 1.73i)21-s + ⋯ |
L(s) = 1 | + (0.612 − 0.353i)2-s + (−0.499 − 0.288i)3-s + (0.249 − 0.433i)4-s − 0.408·6-s + (0.981 + 0.188i)7-s − 0.353i·8-s + (0.166 + 0.288i)9-s + (−0.249 + 0.144i)12-s − 1.10i·13-s + (0.668 − 0.231i)14-s + (−0.125 − 0.216i)16-s + (0.630 + 0.363i)17-s + (0.204 + 0.117i)18-s + (0.229 + 0.397i)19-s + (−0.436 − 0.377i)21-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.324 + 0.946i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.324 + 0.946i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.194091678\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.194091678\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 7 | \( 1 + (-2.59 - 0.5i)T \) |
good | 11 | \( 1 + (-5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + 4iT - 13T^{2} \) |
| 17 | \( 1 + (-2.59 - 1.5i)T + (8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (-1 - 1.73i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (-2.59 + 1.5i)T + (11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + 29T^{2} \) |
| 31 | \( 1 + (-0.5 + 0.866i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (-8.66 + 5i)T + (18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + 9T + 41T^{2} \) |
| 43 | \( 1 + 10iT - 43T^{2} \) |
| 47 | \( 1 + (-2.59 + 1.5i)T + (23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (-5.19 - 3i)T + (26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (3 - 5.19i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (4 + 6.92i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (3.46 + 2i)T + (33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 - 3T + 71T^{2} \) |
| 73 | \( 1 + (12.1 + 7i)T + (36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (-5.5 - 9.52i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 - 83T^{2} \) |
| 89 | \( 1 + (7.5 + 12.9i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 - 7iT - 97T^{2} \) |
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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
−10.14937329676934317808593881346, −8.880772705071301010243253273706, −7.941073921457383355404347126582, −7.26300212250798156546763381173, −6.01241091453760495191790695370, −5.44584949354032406631092878543, −4.62514114628719464529885456154, −3.47337731662721249504516270820, −2.23697661371650768255272198697, −1.00435969372114764072646419576,
1.43127376069712794799117051870, 2.96087516541886852215690935778, 4.24891457771657156706364606690, 4.81305534544587656852827914885, 5.66307821626685989616473577994, 6.66353989099662570627134765323, 7.40199258354259981729376018092, 8.292809161991609326135254237445, 9.254868760261203995110615866939, 10.12867614450451929162552031991