| L(s) = 1 | + (−0.118 − 0.673i)2-s + (1.43 − 0.524i)4-s + (−0.802 − 0.673i)5-s + (0.113 + 0.0412i)7-s + (−1.20 − 2.09i)8-s + (−0.358 + 0.620i)10-s + (4.16 − 3.49i)11-s + (−0.794 + 4.50i)13-s + (0.0143 − 0.0812i)14-s + (1.08 − 0.907i)16-s + (2.38 − 4.13i)17-s + (0.294 + 0.509i)19-s + (−1.50 − 0.549i)20-s + (−2.84 − 2.38i)22-s + (−7.32 + 2.66i)23-s + ⋯ |
| L(s) = 1 | + (−0.0839 − 0.476i)2-s + (0.719 − 0.262i)4-s + (−0.359 − 0.301i)5-s + (0.0428 + 0.0155i)7-s + (−0.427 − 0.739i)8-s + (−0.113 + 0.196i)10-s + (1.25 − 1.05i)11-s + (−0.220 + 1.24i)13-s + (0.00382 − 0.0217i)14-s + (0.270 − 0.226i)16-s + (0.579 − 1.00i)17-s + (0.0675 + 0.116i)19-s + (−0.337 − 0.122i)20-s + (−0.607 − 0.509i)22-s + (−1.52 + 0.555i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 729 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.116 + 0.993i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 729 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.116 + 0.993i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.10340 - 1.23988i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.10340 - 1.23988i\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + (0.118 + 0.673i)T + (-1.87 + 0.684i)T^{2} \) |
| 5 | \( 1 + (0.802 + 0.673i)T + (0.868 + 4.92i)T^{2} \) |
| 7 | \( 1 + (-0.113 - 0.0412i)T + (5.36 + 4.49i)T^{2} \) |
| 11 | \( 1 + (-4.16 + 3.49i)T + (1.91 - 10.8i)T^{2} \) |
| 13 | \( 1 + (0.794 - 4.50i)T + (-12.2 - 4.44i)T^{2} \) |
| 17 | \( 1 + (-2.38 + 4.13i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (-0.294 - 0.509i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (7.32 - 2.66i)T + (17.6 - 14.7i)T^{2} \) |
| 29 | \( 1 + (0.880 + 4.99i)T + (-27.2 + 9.91i)T^{2} \) |
| 31 | \( 1 + (-8.23 + 2.99i)T + (23.7 - 19.9i)T^{2} \) |
| 37 | \( 1 + (-1.09 + 1.89i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (1.31 - 7.44i)T + (-38.5 - 14.0i)T^{2} \) |
| 43 | \( 1 + (1 - 0.839i)T + (7.46 - 42.3i)T^{2} \) |
| 47 | \( 1 + (-2.27 - 0.826i)T + (36.0 + 30.2i)T^{2} \) |
| 53 | \( 1 - 3.04T + 53T^{2} \) |
| 59 | \( 1 + (-0.0336 - 0.0282i)T + (10.2 + 58.1i)T^{2} \) |
| 61 | \( 1 + (-9.59 - 3.49i)T + (46.7 + 39.2i)T^{2} \) |
| 67 | \( 1 + (0.322 - 1.83i)T + (-62.9 - 22.9i)T^{2} \) |
| 71 | \( 1 + (3.25 - 5.63i)T + (-35.5 - 61.4i)T^{2} \) |
| 73 | \( 1 + (6.11 + 10.5i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (-0.121 - 0.691i)T + (-74.2 + 27.0i)T^{2} \) |
| 83 | \( 1 + (1.17 + 6.67i)T + (-77.9 + 28.3i)T^{2} \) |
| 89 | \( 1 + (-3.42 - 5.93i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (6.91 - 5.80i)T + (16.8 - 95.5i)T^{2} \) |
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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
−10.01101781390596414747386271991, −9.574051063570426087683391035992, −8.520886322075752522206995689993, −7.57205587879261426109483584749, −6.49184240988315502854833727361, −5.95397823547172842588345583898, −4.42964974863425480410509938417, −3.53668750172022430466834951789, −2.24214692403022444161756114451, −0.927255857591062732309210297914,
1.71993521062899748954086536967, 3.09258351734180223961451967150, 4.09059409237896830166057466752, 5.46792663807120536977249553499, 6.39853653814362893390645565891, 7.13950886958184406815143406961, 7.915950702661919477052707389532, 8.625318137016238253904861917678, 9.921567360978443820498486200009, 10.52558525683564270878820393590
