L(s) = 1 | + (−0.866 − 0.5i)2-s + (0.499 + 0.866i)4-s + (−0.656 − 2.13i)5-s + (−2.63 + 0.209i)7-s − 0.999i·8-s + (−0.5 + 2.17i)10-s + (−0.866 − 1.5i)11-s + 2.15i·13-s + (2.38 + 1.13i)14-s + (−0.5 + 0.866i)16-s + (−1.73 + i)17-s + (−2.13 + 3.70i)19-s + (1.52 − 1.63i)20-s + 1.73i·22-s + (3.70 + 2.13i)23-s + ⋯ |
L(s) = 1 | + (−0.612 − 0.353i)2-s + (0.249 + 0.433i)4-s + (−0.293 − 0.955i)5-s + (−0.996 + 0.0791i)7-s − 0.353i·8-s + (−0.158 + 0.689i)10-s + (−0.261 − 0.452i)11-s + 0.596i·13-s + (0.638 + 0.303i)14-s + (−0.125 + 0.216i)16-s + (−0.420 + 0.242i)17-s + (−0.490 + 0.849i)19-s + (0.340 − 0.366i)20-s + 0.369i·22-s + (0.771 + 0.445i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.426 - 0.904i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.426 - 0.904i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.0930069 + 0.146690i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.0930069 + 0.146690i\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 5 | \( 1 + (0.656 + 2.13i)T \) |
| 7 | \( 1 + (2.63 - 0.209i)T \) |
good | 11 | \( 1 + (0.866 + 1.5i)T + (-5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 - 2.15iT - 13T^{2} \) |
| 17 | \( 1 + (1.73 - i)T + (8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (2.13 - 3.70i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (-3.70 - 2.13i)T + (11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + 3.88T + 29T^{2} \) |
| 31 | \( 1 + (-3.63 - 6.30i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (4.13 + 2.38i)T + (18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + 10.8T + 41T^{2} \) |
| 43 | \( 1 + 2.62iT - 43T^{2} \) |
| 47 | \( 1 + (-1.49 - 0.862i)T + (23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (6.53 - 3.77i)T + (26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (-3.67 - 6.36i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (2.27 - 3.94i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (7.54 - 4.35i)T + (33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + 14.8T + 71T^{2} \) |
| 73 | \( 1 + (-10.5 + 6.09i)T + (36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (1.63 - 2.83i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 - 0.725iT - 83T^{2} \) |
| 89 | \( 1 + (0.418 - 0.725i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 - 2.20iT - 97T^{2} \) |
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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
−10.73390575072725399584646684370, −9.961936432540646986353276972236, −8.966981082564943983006170790479, −8.634945961795971464371845052175, −7.48795123551934557848442334929, −6.51899375762912002698359463776, −5.43404688255634428269969502947, −4.14696922028121884147161403429, −3.15588388758986341106540663252, −1.57945332213670858800838023593,
0.11249044755246473655737113755, 2.40691679289665940709920575103, 3.39935442108032349004693878002, 4.83874018209562891514928385686, 6.18603212610769527902577216487, 6.81941225711211422340102277822, 7.53580053729665137217110290735, 8.560002014629362611469955328415, 9.554089204820601885272589911398, 10.23590056342267030976966980803