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 + (−0.866 − 2.5i)7-s − 0.999i·8-s + (0.499 + 0.866i)9-s + (2 − 3.46i)11-s + (0.866 − 0.499i)12-s − i·13-s + (−2 − 1.73i)14-s + (−0.5 − 0.866i)16-s + (−1.73 − i)17-s + (0.866 + 0.499i)18-s + (0.5 + 0.866i)19-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 + (−0.327 − 0.944i)7-s − 0.353i·8-s + (0.166 + 0.288i)9-s + (0.603 − 1.04i)11-s + (0.249 − 0.144i)12-s − 0.277i·13-s + (−0.534 − 0.462i)14-s + (−0.125 − 0.216i)16-s + (−0.420 − 0.242i)17-s + (0.204 + 0.117i)18-s + (0.114 + 0.198i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.208 + 0.978i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.208 + 0.978i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.580649986\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.580649986\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + (0.866 + 2.5i)T \) |
good | 11 | \( 1 + (-2 + 3.46i)T + (-5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + iT - 13T^{2} \) |
| 17 | \( 1 + (1.73 + i)T + (8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (-0.5 - 0.866i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (-1.73 + i)T + (11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + 4T + 29T^{2} \) |
| 31 | \( 1 + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (-2.59 + 1.5i)T + (18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 - 12T + 41T^{2} \) |
| 43 | \( 1 + 8iT - 43T^{2} \) |
| 47 | \( 1 + (5.19 - 3i)T + (23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (-1.73 - i)T + (26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (-3 + 5.19i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (-6.5 - 11.2i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (2.59 + 1.5i)T + (33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 - 16T + 71T^{2} \) |
| 73 | \( 1 + (9.52 + 5.5i)T + (36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (-6.5 - 11.2i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 - 6iT - 83T^{2} \) |
| 89 | \( 1 + (1 + 1.73i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 - 17iT - 97T^{2} \) |
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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
−9.738197517721707213877801081541, −9.107193785048249188554082585612, −8.079563813658932758883573987157, −7.15113659779806067395518645274, −6.28182236583458764167666509662, −5.29963376985188892629310228706, −4.12130299811634781680125576415, −3.59848743668428174027061162901, −2.53299316946435144913347081046, −0.934352102489432525779558567487,
1.84262058371971812591948548075, 2.79877002595630844501861641770, 3.93424177375578460057053015644, 4.85144107346652865181770554042, 5.94048818279821578484593436124, 6.69322319013603972494551943242, 7.45096195402026379387795396774, 8.393658455275971895771611706086, 9.252171221010297492082099673912, 9.755364049381926908279821304770