| 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
−10.49963460249356692855883635379, −9.320339158473860774078670116667, −8.405576926674857487091439247152, −7.68677514393026789303463207582, −6.93512127881762764066274837415, −5.54461408418587548890935358456, −4.87479508401915487543307573503, −4.47704829285234417780298268669, −2.16751927879456347556732961206, −1.22839831811550724511080284676,
2.03348395667039946694793724355, 2.52344235855074283530379470655, 3.88630429422347408250884521284, 4.85971666713314271175686989245, 6.08821412850620867207634596938, 7.12327577863510223289673362517, 7.87263873709012204109717925480, 8.501933023308981659874864071222, 9.969253443021464633464070253267, 10.88485023717967777329166192935
