L(s) = 1 | + (−1.36 − 2.36i)2-s + (−0.583 − 1.63i)3-s + (−2.73 + 4.74i)4-s + (−3.06 + 3.61i)6-s + (−1.39 + 2.24i)7-s + 9.49·8-s + (−2.31 + 1.90i)9-s + (1.79 + 1.03i)11-s + (9.32 + 1.69i)12-s + 2.04·13-s + (7.22 + 0.226i)14-s + (−7.50 − 13.0i)16-s + (−1.82 − 1.05i)17-s + (7.67 + 2.88i)18-s + (2.41 − 1.39i)19-s + ⋯ |
L(s) = 1 | + (−0.966 − 1.67i)2-s + (−0.336 − 0.941i)3-s + (−1.36 + 2.37i)4-s + (−1.25 + 1.47i)6-s + (−0.526 + 0.849i)7-s + 3.35·8-s + (−0.773 + 0.634i)9-s + (0.540 + 0.312i)11-s + (2.69 + 0.490i)12-s + 0.566·13-s + (1.93 + 0.0604i)14-s + (−1.87 − 3.25i)16-s + (−0.441 − 0.255i)17-s + (1.80 + 0.681i)18-s + (0.553 − 0.319i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.144 + 0.989i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.144 + 0.989i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.415331 - 0.480146i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.415331 - 0.480146i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (0.583 + 1.63i)T \) |
| 5 | \( 1 \) |
| 7 | \( 1 + (1.39 - 2.24i)T \) |
good | 2 | \( 1 + (1.36 + 2.36i)T + (-1 + 1.73i)T^{2} \) |
| 11 | \( 1 + (-1.79 - 1.03i)T + (5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 - 2.04T + 13T^{2} \) |
| 17 | \( 1 + (1.82 + 1.05i)T + (8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (-2.41 + 1.39i)T + (9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (0.888 + 1.53i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 - 2.79iT - 29T^{2} \) |
| 31 | \( 1 + (-6.04 - 3.49i)T + (15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (-5.25 + 3.03i)T + (18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 - 11.7T + 41T^{2} \) |
| 43 | \( 1 - 4.37iT - 43T^{2} \) |
| 47 | \( 1 + (2.55 - 1.47i)T + (23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (3.09 - 5.36i)T + (-26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (-1.44 + 2.49i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (6.30 - 3.63i)T + (30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (-6.41 - 3.70i)T + (33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 - 3.80iT - 71T^{2} \) |
| 73 | \( 1 + (-3.35 + 5.81i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (2.95 + 5.12i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 - 5.96iT - 83T^{2} \) |
| 89 | \( 1 + (-6.20 - 10.7i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 - 9.97T + 97T^{2} \) |
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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
−10.87241249902553624465961860791, −9.708346836853536779840971637587, −9.029518140997705280095300461744, −8.290218591155529749414952260545, −7.29396873383228457577604214219, −6.15386548129071287138805818725, −4.60623854197196978224719546578, −3.13488462395281047570309142178, −2.26898287045353166982533124726, −0.993390110816634887919415843978,
0.75304655652373080329277324053, 3.82397131689017592004895580946, 4.73047241638403866040787124816, 6.02259391211602027751969607457, 6.35517439005821317356934608638, 7.54166996792319446355490179848, 8.413098404021352198105406436959, 9.348425107689762378246878660193, 9.852035710946998062149934058678, 10.65089241247194246537990241821