| 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
−10.32939520127320655023758785490, −9.060225514620763825003232707957, −8.571463931179296695472960794429, −8.248606974161739232919703641222, −6.86201785760825943631615242785, −5.64319406675265306078702217849, −4.93970032785249773356216957193, −4.01630927307282002312722771103, −2.50092236186192429771682879768, −1.11614418196031088320398159626,
1.65146684811524215896731774816, 2.27811097801095353672682544306, 4.28318983513271977017497067031, 4.96033602780639362141814238324, 6.11787201542077577789886597084, 6.74481824968623543299594869407, 7.81672263961567302634094685309, 8.969946222061546948951968473961, 9.754310418794973086448746048125, 10.42160940351647462184973889374
