L(s) = 1 | + (−0.5 + 0.866i)2-s + (−0.866 + 0.5i)3-s + (−0.499 − 0.866i)4-s + (1.83 − 1.28i)5-s − 0.999i·6-s + (−3.31 + 1.91i)7-s + 0.999·8-s + (0.499 − 0.866i)9-s + (0.196 + 2.22i)10-s + 1.67·11-s + (0.866 + 0.499i)12-s + (1.56 + 2.70i)13-s − 3.82i·14-s + (−0.943 + 2.02i)15-s + (−0.5 + 0.866i)16-s + (2.75 − 4.78i)17-s + ⋯ |
L(s) = 1 | + (−0.353 + 0.612i)2-s + (−0.499 + 0.288i)3-s + (−0.249 − 0.433i)4-s + (0.818 − 0.574i)5-s − 0.408i·6-s + (−1.25 + 0.722i)7-s + 0.353·8-s + (0.166 − 0.288i)9-s + (0.0622 + 0.704i)10-s + 0.505·11-s + (0.249 + 0.144i)12-s + (0.432 + 0.749i)13-s − 1.02i·14-s + (−0.243 + 0.523i)15-s + (−0.125 + 0.216i)16-s + (0.669 − 1.15i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0305 - 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0305 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.071565819\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.071565819\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.5 - 0.866i)T \) |
| 3 | \( 1 + (0.866 - 0.5i)T \) |
| 5 | \( 1 + (-1.83 + 1.28i)T \) |
| 37 | \( 1 + (1.06 + 5.98i)T \) |
good | 7 | \( 1 + (3.31 - 1.91i)T + (3.5 - 6.06i)T^{2} \) |
| 11 | \( 1 - 1.67T + 11T^{2} \) |
| 13 | \( 1 + (-1.56 - 2.70i)T + (-6.5 + 11.2i)T^{2} \) |
| 17 | \( 1 + (-2.75 + 4.78i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (1.27 - 0.737i)T + (9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + 1.21T + 23T^{2} \) |
| 29 | \( 1 - 2.97iT - 29T^{2} \) |
| 31 | \( 1 - 5.65iT - 31T^{2} \) |
| 41 | \( 1 + (-4.89 - 8.47i)T + (-20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 - 11.7T + 43T^{2} \) |
| 47 | \( 1 - 11.2iT - 47T^{2} \) |
| 53 | \( 1 + (-2.49 - 1.44i)T + (26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (-5.20 - 3.00i)T + (29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (5.55 - 3.20i)T + (30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (-0.933 + 0.539i)T + (33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + (-0.807 - 1.39i)T + (-35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 - 12.7iT - 73T^{2} \) |
| 79 | \( 1 + (1.38 - 0.796i)T + (39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (-3.79 - 2.19i)T + (41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 + (7.36 + 4.24i)T + (44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 - 16.8T + 97T^{2} \) |
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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
−9.700354579244243847228755201834, −9.250420443666061854304100845325, −8.788170876626049585298235819526, −7.40021057884982036345804130575, −6.42641572225598171279768400265, −5.98970134159782658954947187443, −5.18638632917573845208590983173, −4.12080227901542686470321815700, −2.71118546888481468900615060226, −1.11855895720703199040885521618,
0.68941841414823323499711563240, 2.06365206570027762750675995924, 3.29332241480702939516451052185, 4.04494662105179991962197161401, 5.68416311015237762999088441754, 6.24750127637064026665248154649, 7.05814828081797850542734591178, 7.969227885370889714196274498683, 9.106719157319636956035820662603, 9.914384140321169073594748223844