L(s) = 1 | + 1.72·2-s + 0.981·4-s + (−1.75 − 3.04i)5-s − 1.75·8-s + (−3.03 − 5.25i)10-s + (−3.04 + 5.27i)11-s + (−0.560 + 0.970i)13-s − 4.99·16-s + (−0.601 − 1.04i)17-s + (−1.10 + 1.90i)19-s + (−1.72 − 2.98i)20-s + (−5.25 + 9.10i)22-s + (−0.636 − 1.10i)23-s + (−3.66 + 6.35i)25-s + (−0.967 + 1.67i)26-s + ⋯ |
L(s) = 1 | + 1.22·2-s + 0.490·4-s + (−0.785 − 1.36i)5-s − 0.621·8-s + (−0.958 − 1.66i)10-s + (−0.918 + 1.59i)11-s + (−0.155 + 0.269i)13-s − 1.24·16-s + (−0.146 − 0.252i)17-s + (−0.252 + 0.438i)19-s + (−0.385 − 0.667i)20-s + (−1.12 + 1.94i)22-s + (−0.132 − 0.229i)23-s + (−0.733 + 1.27i)25-s + (−0.189 + 0.328i)26-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.835 - 0.549i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.835 - 0.549i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.2111213216\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.2111213216\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 7 | \( 1 \) |
good | 2 | \( 1 - 1.72T + 2T^{2} \) |
| 5 | \( 1 + (1.75 + 3.04i)T + (-2.5 + 4.33i)T^{2} \) |
| 11 | \( 1 + (3.04 - 5.27i)T + (-5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (0.560 - 0.970i)T + (-6.5 - 11.2i)T^{2} \) |
| 17 | \( 1 + (0.601 + 1.04i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (1.10 - 1.90i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (0.636 + 1.10i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (-3.10 - 5.37i)T + (-14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 - 0.188T + 31T^{2} \) |
| 37 | \( 1 + (1.78 - 3.09i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (-1.68 + 2.91i)T + (-20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (1.90 + 3.29i)T + (-21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + 5.72T + 47T^{2} \) |
| 53 | \( 1 + (4.16 + 7.22i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + 11.2T + 59T^{2} \) |
| 61 | \( 1 + 12.0T + 61T^{2} \) |
| 67 | \( 1 + 7.91T + 67T^{2} \) |
| 71 | \( 1 - 12.2T + 71T^{2} \) |
| 73 | \( 1 + (2.65 + 4.60i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 - 9.21T + 79T^{2} \) |
| 83 | \( 1 + (-0.624 - 1.08i)T + (-41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 + (2.77 - 4.79i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (8.24 + 14.2i)T + (-48.5 + 84.0i)T^{2} \) |
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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
−9.881343789496044031549164441636, −9.097297338287137490547796904268, −8.298664573183191520594786489386, −7.48347221833569760597014903186, −6.53727533982237218900097338104, −5.30998204016749638064800625587, −4.75763370590357805447036205626, −4.33642505227542743852678940990, −3.20448973869464947991784868845, −1.86078842225513506993396819489,
0.05418274746174763624479014916, 2.70123167387637657470613329330, 3.13168905755224842472335472302, 3.99550234732566368684991070296, 4.98878398495878168746351878036, 6.04177117970317296280839463020, 6.45784922039329441574572395568, 7.63834392861864040805400777836, 8.215318074737844882809539312889, 9.304859991615977308799031445101