L(s) = 1 | + (−1.34 + 0.776i)2-s + (0.204 − 0.355i)4-s + (−1.88 + 1.19i)5-s + (−0.478 − 2.60i)7-s − 2.46i·8-s + (1.61 − 3.07i)10-s + (2.21 − 3.83i)11-s + 1.73i·13-s + (2.66 + 3.12i)14-s + (2.32 + 4.02i)16-s + (2.36 + 1.36i)17-s + (−0.152 − 0.264i)19-s + (0.0376 + 0.915i)20-s + 6.87i·22-s + (6.08 − 3.51i)23-s + ⋯ |
L(s) = 1 | + (−0.950 + 0.548i)2-s + (0.102 − 0.177i)4-s + (−0.844 + 0.535i)5-s + (−0.180 − 0.983i)7-s − 0.872i·8-s + (0.509 − 0.972i)10-s + (0.667 − 1.15i)11-s + 0.480i·13-s + (0.711 + 0.835i)14-s + (0.581 + 1.00i)16-s + (0.573 + 0.331i)17-s + (−0.0350 − 0.0606i)19-s + (0.00841 + 0.204i)20-s + 1.46i·22-s + (1.26 − 0.732i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.934 + 0.356i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.934 + 0.356i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.573865 - 0.105645i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.573865 - 0.105645i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 5 | \( 1 + (1.88 - 1.19i)T \) |
| 7 | \( 1 + (0.478 + 2.60i)T \) |
good | 2 | \( 1 + (1.34 - 0.776i)T + (1 - 1.73i)T^{2} \) |
| 11 | \( 1 + (-2.21 + 3.83i)T + (-5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 - 1.73iT - 13T^{2} \) |
| 17 | \( 1 + (-2.36 - 1.36i)T + (8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (0.152 + 0.264i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (-6.08 + 3.51i)T + (11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + 7.79T + 29T^{2} \) |
| 31 | \( 1 + (-2.64 + 4.57i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (-3.18 + 1.83i)T + (18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 - 6.71T + 41T^{2} \) |
| 43 | \( 1 + 9.71iT - 43T^{2} \) |
| 47 | \( 1 + (-1.57 + 0.908i)T + (23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (1.48 + 0.857i)T + (26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (-0.571 + 0.989i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (4.77 + 8.27i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (7.26 + 4.19i)T + (33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + 10.2T + 71T^{2} \) |
| 73 | \( 1 + (-11.1 - 6.41i)T + (36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (-3.35 - 5.81i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + 5.09iT - 83T^{2} \) |
| 89 | \( 1 + (2.03 + 3.52i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 - 2.87iT - 97T^{2} \) |
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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
−11.25603905095156715573640218643, −10.68977326710736130904180424983, −9.539669606972393088823823723778, −8.670145978299532175866671887061, −7.74202829023759922154098671568, −7.05734507838956761402769648862, −6.18091762005368408471840137084, −4.17211301925413647405630253334, −3.40423287425680787396726365236, −0.66649797788646870767222291466,
1.36960465278444041911343575474, 2.98983776816962518657238932961, 4.64417992514253459008819984240, 5.64458848648665916608838013206, 7.28671601497938325057510193200, 8.136514549004675479920802036537, 9.257818521513981396243879110888, 9.458323813592195232204565507929, 10.80080753060879421706925289427, 11.68065430973539543878707101994