| L(s) = 1 | + (−0.103 − 0.0680i)2-s + (−0.786 − 1.82i)4-s + (2.31 + 2.45i)5-s + (3.71 − 0.434i)7-s + (−0.0857 + 0.486i)8-s + (−0.0724 − 0.411i)10-s + (1.30 + 4.37i)11-s + (0.0739 + 1.26i)13-s + (−0.414 − 0.207i)14-s + (−2.68 + 2.84i)16-s + (−1.54 − 0.563i)17-s + (−4.14 + 1.50i)19-s + (2.64 − 6.14i)20-s + (0.162 − 0.541i)22-s + (1.06 + 0.123i)23-s + ⋯ |
| L(s) = 1 | + (−0.0731 − 0.0481i)2-s + (−0.393 − 0.911i)4-s + (1.03 + 1.09i)5-s + (1.40 − 0.164i)7-s + (−0.0303 + 0.171i)8-s + (−0.0229 − 0.130i)10-s + (0.394 + 1.31i)11-s + (0.0205 + 0.351i)13-s + (−0.110 − 0.0555i)14-s + (−0.670 + 0.710i)16-s + (−0.375 − 0.136i)17-s + (−0.950 + 0.346i)19-s + (0.592 − 1.37i)20-s + (0.0345 − 0.115i)22-s + (0.221 + 0.0258i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 729 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.921 - 0.387i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 729 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.921 - 0.387i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.78069 + 0.359052i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.78069 + 0.359052i\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + (0.103 + 0.0680i)T + (0.792 + 1.83i)T^{2} \) |
| 5 | \( 1 + (-2.31 - 2.45i)T + (-0.290 + 4.99i)T^{2} \) |
| 7 | \( 1 + (-3.71 + 0.434i)T + (6.81 - 1.61i)T^{2} \) |
| 11 | \( 1 + (-1.30 - 4.37i)T + (-9.19 + 6.04i)T^{2} \) |
| 13 | \( 1 + (-0.0739 - 1.26i)T + (-12.9 + 1.50i)T^{2} \) |
| 17 | \( 1 + (1.54 + 0.563i)T + (13.0 + 10.9i)T^{2} \) |
| 19 | \( 1 + (4.14 - 1.50i)T + (14.5 - 12.2i)T^{2} \) |
| 23 | \( 1 + (-1.06 - 0.123i)T + (22.3 + 5.30i)T^{2} \) |
| 29 | \( 1 + (-1.61 + 0.809i)T + (17.3 - 23.2i)T^{2} \) |
| 31 | \( 1 + (-4.05 + 5.44i)T + (-8.89 - 29.6i)T^{2} \) |
| 37 | \( 1 + (2.46 - 2.06i)T + (6.42 - 36.4i)T^{2} \) |
| 41 | \( 1 + (-0.842 + 0.553i)T + (16.2 - 37.6i)T^{2} \) |
| 43 | \( 1 + (2.24 + 0.531i)T + (38.4 + 19.2i)T^{2} \) |
| 47 | \( 1 + (-5.68 - 7.64i)T + (-13.4 + 45.0i)T^{2} \) |
| 53 | \( 1 + (-1.43 - 2.49i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (-3.05 + 10.2i)T + (-49.2 - 32.4i)T^{2} \) |
| 61 | \( 1 + (-2.10 + 4.87i)T + (-41.8 - 44.3i)T^{2} \) |
| 67 | \( 1 + (2.71 + 1.36i)T + (40.0 + 53.7i)T^{2} \) |
| 71 | \( 1 + (0.346 + 1.96i)T + (-66.7 + 24.2i)T^{2} \) |
| 73 | \( 1 + (-2.80 + 15.8i)T + (-68.5 - 24.9i)T^{2} \) |
| 79 | \( 1 + (-4.69 - 3.08i)T + (31.2 + 72.5i)T^{2} \) |
| 83 | \( 1 + (10.0 + 6.62i)T + (32.8 + 76.2i)T^{2} \) |
| 89 | \( 1 + (-1.69 + 9.63i)T + (-83.6 - 30.4i)T^{2} \) |
| 97 | \( 1 + (-6.44 + 6.83i)T + (-5.64 - 96.8i)T^{2} \) |
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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
−10.42160940351647462184973889374, −9.754310418794973086448746048125, −8.969946222061546948951968473961, −7.81672263961567302634094685309, −6.74481824968623543299594869407, −6.11787201542077577789886597084, −4.96033602780639362141814238324, −4.28318983513271977017497067031, −2.27811097801095353672682544306, −1.65146684811524215896731774816,
1.11614418196031088320398159626, 2.50092236186192429771682879768, 4.01630927307282002312722771103, 4.93970032785249773356216957193, 5.64319406675265306078702217849, 6.86201785760825943631615242785, 8.248606974161739232919703641222, 8.571463931179296695472960794429, 9.060225514620763825003232707957, 10.32939520127320655023758785490
