| L(s) = 1 | + (−0.123 − 2.11i)2-s + (−2.45 + 0.287i)4-s + (−3.78 − 0.897i)5-s + (−1.99 + 2.67i)7-s + (0.175 + 0.992i)8-s + (−1.43 + 8.11i)10-s + (0.519 + 0.550i)11-s + (2.22 − 1.11i)13-s + (5.89 + 3.87i)14-s + (−2.74 + 0.650i)16-s + (0.700 − 0.255i)17-s + (4.21 + 1.53i)19-s + (9.57 + 1.11i)20-s + (1.09 − 1.16i)22-s + (1.36 + 1.83i)23-s + ⋯ |
| L(s) = 1 | + (−0.0869 − 1.49i)2-s + (−1.22 + 0.143i)4-s + (−1.69 − 0.401i)5-s + (−0.752 + 1.01i)7-s + (0.0618 + 0.350i)8-s + (−0.452 + 2.56i)10-s + (0.156 + 0.165i)11-s + (0.616 − 0.309i)13-s + (1.57 + 1.03i)14-s + (−0.686 + 0.162i)16-s + (0.170 − 0.0618i)17-s + (0.967 + 0.352i)19-s + (2.14 + 0.250i)20-s + (0.234 − 0.248i)22-s + (0.284 + 0.382i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 729 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.978 + 0.207i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 729 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.978 + 0.207i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.583479 - 0.0610647i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.583479 - 0.0610647i\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + (0.123 + 2.11i)T + (-1.98 + 0.232i)T^{2} \) |
| 5 | \( 1 + (3.78 + 0.897i)T + (4.46 + 2.24i)T^{2} \) |
| 7 | \( 1 + (1.99 - 2.67i)T + (-2.00 - 6.70i)T^{2} \) |
| 11 | \( 1 + (-0.519 - 0.550i)T + (-0.639 + 10.9i)T^{2} \) |
| 13 | \( 1 + (-2.22 + 1.11i)T + (7.76 - 10.4i)T^{2} \) |
| 17 | \( 1 + (-0.700 + 0.255i)T + (13.0 - 10.9i)T^{2} \) |
| 19 | \( 1 + (-4.21 - 1.53i)T + (14.5 + 12.2i)T^{2} \) |
| 23 | \( 1 + (-1.36 - 1.83i)T + (-6.59 + 22.0i)T^{2} \) |
| 29 | \( 1 + (-0.387 + 0.254i)T + (11.4 - 26.6i)T^{2} \) |
| 31 | \( 1 + (1.56 - 3.61i)T + (-21.2 - 22.5i)T^{2} \) |
| 37 | \( 1 + (3.64 + 3.05i)T + (6.42 + 36.4i)T^{2} \) |
| 41 | \( 1 + (-0.284 + 4.88i)T + (-40.7 - 4.75i)T^{2} \) |
| 43 | \( 1 + (1.70 - 5.69i)T + (-35.9 - 23.6i)T^{2} \) |
| 47 | \( 1 + (-4.26 - 9.88i)T + (-32.2 + 34.1i)T^{2} \) |
| 53 | \( 1 + (5.75 - 9.96i)T + (-26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (2.84 - 3.01i)T + (-3.43 - 58.9i)T^{2} \) |
| 61 | \( 1 + (0.265 + 0.0310i)T + (59.3 + 14.0i)T^{2} \) |
| 67 | \( 1 + (-1.60 - 1.05i)T + (26.5 + 61.5i)T^{2} \) |
| 71 | \( 1 + (1.17 - 6.65i)T + (-66.7 - 24.2i)T^{2} \) |
| 73 | \( 1 + (-1.37 - 7.81i)T + (-68.5 + 24.9i)T^{2} \) |
| 79 | \( 1 + (0.250 + 4.30i)T + (-78.4 + 9.17i)T^{2} \) |
| 83 | \( 1 + (0.148 + 2.54i)T + (-82.4 + 9.63i)T^{2} \) |
| 89 | \( 1 + (-0.935 - 5.30i)T + (-83.6 + 30.4i)T^{2} \) |
| 97 | \( 1 + (-9.23 + 2.18i)T + (86.6 - 43.5i)T^{2} \) |
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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
−10.64947165441924347344036750246, −9.476700889869527672175771238045, −8.969019911930221995865417396245, −8.050653487284427313900713200464, −7.05078166230536044782048893594, −5.67321104161612825137681634961, −4.43811119780649473813930157798, −3.50564232964777110654377933410, −2.91182593599726494898563950629, −1.16952168798684932505419871870,
0.37349596429082385154812415692, 3.31234578916321907748880675740, 4.02511324128793227069623281673, 5.08878464586075283833174223423, 6.47455148914550284281536387331, 6.94946435612541745570967140952, 7.64730276833802474235432406824, 8.312243451566023799541317317886, 9.227786362539463001092766527946, 10.39101453137973942269515897941
