| L(s) = 1 | + (0.311 − 0.723i)2-s + (0.946 + 1.00i)4-s + (0.161 + 2.78i)5-s + (4.84 + 1.14i)7-s + (2.50 − 0.910i)8-s + (2.06 + 0.750i)10-s + (−1.45 − 0.954i)11-s + (−2.15 − 0.252i)13-s + (2.34 − 3.14i)14-s + (−0.0385 + 0.662i)16-s + (−3.39 + 2.84i)17-s + (−1.63 − 1.37i)19-s + (−2.63 + 2.79i)20-s + (−1.14 + 0.751i)22-s + (−0.659 + 0.156i)23-s + ⋯ |
| L(s) = 1 | + (0.220 − 0.511i)2-s + (0.473 + 0.501i)4-s + (0.0724 + 1.24i)5-s + (1.83 + 0.434i)7-s + (0.884 − 0.321i)8-s + (0.651 + 0.237i)10-s + (−0.437 − 0.287i)11-s + (−0.598 − 0.0699i)13-s + (0.626 − 0.841i)14-s + (−0.00964 + 0.165i)16-s + (−0.823 + 0.690i)17-s + (−0.374 − 0.314i)19-s + (−0.589 + 0.625i)20-s + (−0.243 + 0.160i)22-s + (−0.137 + 0.0326i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 729 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.816 - 0.576i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 729 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.816 - 0.576i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.17578 + 0.690818i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.17578 + 0.690818i\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + (-0.311 + 0.723i)T + (-1.37 - 1.45i)T^{2} \) |
| 5 | \( 1 + (-0.161 - 2.78i)T + (-4.96 + 0.580i)T^{2} \) |
| 7 | \( 1 + (-4.84 - 1.14i)T + (6.25 + 3.14i)T^{2} \) |
| 11 | \( 1 + (1.45 + 0.954i)T + (4.35 + 10.1i)T^{2} \) |
| 13 | \( 1 + (2.15 + 0.252i)T + (12.6 + 2.99i)T^{2} \) |
| 17 | \( 1 + (3.39 - 2.84i)T + (2.95 - 16.7i)T^{2} \) |
| 19 | \( 1 + (1.63 + 1.37i)T + (3.29 + 18.7i)T^{2} \) |
| 23 | \( 1 + (0.659 - 0.156i)T + (20.5 - 10.3i)T^{2} \) |
| 29 | \( 1 + (3.43 + 4.61i)T + (-8.31 + 27.7i)T^{2} \) |
| 31 | \( 1 + (-1.68 + 5.63i)T + (-25.9 - 17.0i)T^{2} \) |
| 37 | \( 1 + (-0.131 - 0.747i)T + (-34.7 + 12.6i)T^{2} \) |
| 41 | \( 1 + (-0.0489 - 0.113i)T + (-28.1 + 29.8i)T^{2} \) |
| 43 | \( 1 + (-2.27 + 1.14i)T + (25.6 - 34.4i)T^{2} \) |
| 47 | \( 1 + (0.487 + 1.62i)T + (-39.2 + 25.8i)T^{2} \) |
| 53 | \( 1 + (5.02 + 8.69i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (-9.71 + 6.39i)T + (23.3 - 54.1i)T^{2} \) |
| 61 | \( 1 + (5.43 - 5.76i)T + (-3.54 - 60.8i)T^{2} \) |
| 67 | \( 1 + (0.277 - 0.373i)T + (-19.2 - 64.1i)T^{2} \) |
| 71 | \( 1 + (-11.4 - 4.17i)T + (54.3 + 45.6i)T^{2} \) |
| 73 | \( 1 + (2.01 - 0.732i)T + (55.9 - 46.9i)T^{2} \) |
| 79 | \( 1 + (-2.77 + 6.43i)T + (-54.2 - 57.4i)T^{2} \) |
| 83 | \( 1 + (1.31 - 3.03i)T + (-56.9 - 60.3i)T^{2} \) |
| 89 | \( 1 + (4.72 - 1.71i)T + (68.1 - 57.2i)T^{2} \) |
| 97 | \( 1 + (-0.436 + 7.49i)T + (-96.3 - 11.2i)T^{2} \) |
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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
−10.88485023717967777329166192935, −9.969253443021464633464070253267, −8.501933023308981659874864071222, −7.87263873709012204109717925480, −7.12327577863510223289673362517, −6.08821412850620867207634596938, −4.85971666713314271175686989245, −3.88630429422347408250884521284, −2.52344235855074283530379470655, −2.03348395667039946694793724355,
1.22839831811550724511080284676, 2.16751927879456347556732961206, 4.47704829285234417780298268669, 4.87479508401915487543307573503, 5.54461408418587548890935358456, 6.93512127881762764066274837415, 7.68677514393026789303463207582, 8.405576926674857487091439247152, 9.320339158473860774078670116667, 10.49963460249356692855883635379
