| L(s) = 1 | + (1.68 − 1.10i)2-s + (0.809 − 1.87i)4-s + (−2.40 + 2.54i)5-s + (−2.90 − 0.339i)7-s + (−0.0156 − 0.0887i)8-s + (−1.22 + 6.93i)10-s + (−0.442 + 1.47i)11-s + (−0.123 + 2.12i)13-s + (−5.26 + 2.64i)14-s + (2.68 + 2.84i)16-s + (−4.47 + 1.62i)17-s + (−5.79 − 2.10i)19-s + (2.83 + 6.57i)20-s + (0.890 + 2.97i)22-s + (1.94 − 0.226i)23-s + ⋯ |
| L(s) = 1 | + (1.18 − 0.781i)2-s + (0.404 − 0.938i)4-s + (−1.07 + 1.13i)5-s + (−1.09 − 0.128i)7-s + (−0.00553 − 0.0313i)8-s + (−0.386 + 2.19i)10-s + (−0.133 + 0.445i)11-s + (−0.0342 + 0.588i)13-s + (−1.40 + 0.706i)14-s + (0.670 + 0.710i)16-s + (−1.08 + 0.394i)17-s + (−1.32 − 0.483i)19-s + (0.634 + 1.47i)20-s + (0.189 + 0.633i)22-s + (0.404 − 0.0473i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 729 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0244 - 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 729 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0244 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.881065 + 0.859799i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.881065 + 0.859799i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| good | 2 | \( 1 + (-1.68 + 1.10i)T + (0.792 - 1.83i)T^{2} \) |
| 5 | \( 1 + (2.40 - 2.54i)T + (-0.290 - 4.99i)T^{2} \) |
| 7 | \( 1 + (2.90 + 0.339i)T + (6.81 + 1.61i)T^{2} \) |
| 11 | \( 1 + (0.442 - 1.47i)T + (-9.19 - 6.04i)T^{2} \) |
| 13 | \( 1 + (0.123 - 2.12i)T + (-12.9 - 1.50i)T^{2} \) |
| 17 | \( 1 + (4.47 - 1.62i)T + (13.0 - 10.9i)T^{2} \) |
| 19 | \( 1 + (5.79 + 2.10i)T + (14.5 + 12.2i)T^{2} \) |
| 23 | \( 1 + (-1.94 + 0.226i)T + (22.3 - 5.30i)T^{2} \) |
| 29 | \( 1 + (-5.17 - 2.60i)T + (17.3 + 23.2i)T^{2} \) |
| 31 | \( 1 + (0.550 + 0.739i)T + (-8.89 + 29.6i)T^{2} \) |
| 37 | \( 1 + (-2.61 - 2.19i)T + (6.42 + 36.4i)T^{2} \) |
| 41 | \( 1 + (-7.40 - 4.86i)T + (16.2 + 37.6i)T^{2} \) |
| 43 | \( 1 + (4.29 - 1.01i)T + (38.4 - 19.2i)T^{2} \) |
| 47 | \( 1 + (-3.76 + 5.05i)T + (-13.4 - 45.0i)T^{2} \) |
| 53 | \( 1 + (4.19 - 7.27i)T + (-26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (-0.648 - 2.16i)T + (-49.2 + 32.4i)T^{2} \) |
| 61 | \( 1 + (5.46 + 12.6i)T + (-41.8 + 44.3i)T^{2} \) |
| 67 | \( 1 + (-1.80 + 0.905i)T + (40.0 - 53.7i)T^{2} \) |
| 71 | \( 1 + (-0.0422 + 0.239i)T + (-66.7 - 24.2i)T^{2} \) |
| 73 | \( 1 + (0.806 + 4.57i)T + (-68.5 + 24.9i)T^{2} \) |
| 79 | \( 1 + (-2.65 + 1.74i)T + (31.2 - 72.5i)T^{2} \) |
| 83 | \( 1 + (12.3 - 8.09i)T + (32.8 - 76.2i)T^{2} \) |
| 89 | \( 1 + (-1.79 - 10.2i)T + (-83.6 + 30.4i)T^{2} \) |
| 97 | \( 1 + (-9.08 - 9.63i)T + (-5.64 + 96.8i)T^{2} \) |
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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
−10.91114013414980309681683445677, −10.21902633589662167150305110844, −8.993476198681994401652791162582, −7.87324248440082492210291101616, −6.67975838244408177151880622183, −6.39613638188669010907686937136, −4.67605485306113404983230366292, −4.04704027609034899851638632268, −3.13790331768941663650777396363, −2.34581914292635146916839901825,
0.41036061828102974761335437108, 2.95665785903044407451368388494, 4.04334823286648257293182282957, 4.60548501816864167291748672994, 5.68591141063846265325570541435, 6.44707415788726898496775996486, 7.38858966574536610992238136590, 8.333570955427339069730755424211, 9.045495588536345956385944021527, 10.18110443014564009857953055420
