| L(s) = 1 | + (−1.22 − 1.32i)2-s + (0.168 + 1.72i)3-s + (−0.0944 + 1.25i)4-s + (0.684 + 0.858i)5-s + (2.07 − 2.34i)6-s + (1.06 − 2.42i)7-s + (−1.04 + 0.829i)8-s + (−2.94 + 0.582i)9-s + (0.295 − 1.96i)10-s + (4.90 − 1.11i)11-s + (−2.18 + 0.0500i)12-s + (−2.31 − 2.49i)13-s + (−4.51 + 1.56i)14-s + (−1.36 + 1.32i)15-s + (4.87 + 0.734i)16-s + (0.0541 + 0.722i)17-s + ⋯ |
| L(s) = 1 | + (−0.868 − 0.936i)2-s + (0.0975 + 0.995i)3-s + (−0.0472 + 0.629i)4-s + (0.306 + 0.383i)5-s + (0.847 − 0.955i)6-s + (0.402 − 0.915i)7-s + (−0.367 + 0.293i)8-s + (−0.980 + 0.194i)9-s + (0.0934 − 0.620i)10-s + (1.47 − 0.337i)11-s + (−0.631 + 0.0144i)12-s + (−0.642 − 0.692i)13-s + (−1.20 + 0.418i)14-s + (−0.352 + 0.342i)15-s + (1.21 + 0.183i)16-s + (0.0131 + 0.175i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.857 + 0.514i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.857 + 0.514i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.981468 - 0.272100i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.981468 - 0.272100i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-0.168 - 1.72i)T \) |
| 7 | \( 1 + (-1.06 + 2.42i)T \) |
| good | 2 | \( 1 + (1.22 + 1.32i)T + (-0.149 + 1.99i)T^{2} \) |
| 5 | \( 1 + (-0.684 - 0.858i)T + (-1.11 + 4.87i)T^{2} \) |
| 11 | \( 1 + (-4.90 + 1.11i)T + (9.91 - 4.77i)T^{2} \) |
| 13 | \( 1 + (2.31 + 2.49i)T + (-0.971 + 12.9i)T^{2} \) |
| 17 | \( 1 + (-0.0541 - 0.722i)T + (-16.8 + 2.53i)T^{2} \) |
| 19 | \( 1 + (-3.19 - 1.84i)T + (9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (-1.87 - 3.89i)T + (-14.3 + 17.9i)T^{2} \) |
| 29 | \( 1 + (-4.96 - 7.27i)T + (-10.5 + 26.9i)T^{2} \) |
| 31 | \( 1 + (-5.05 - 2.91i)T + (15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (-8.62 + 5.87i)T + (13.5 - 34.4i)T^{2} \) |
| 41 | \( 1 + (0.00356 - 0.00907i)T + (-30.0 - 27.8i)T^{2} \) |
| 43 | \( 1 + (1.60 + 4.09i)T + (-31.5 + 29.2i)T^{2} \) |
| 47 | \( 1 + (-0.570 + 0.529i)T + (3.51 - 46.8i)T^{2} \) |
| 53 | \( 1 + (0.0965 - 0.141i)T + (-19.3 - 49.3i)T^{2} \) |
| 59 | \( 1 + (-1.11 - 2.83i)T + (-43.2 + 40.1i)T^{2} \) |
| 61 | \( 1 + (13.4 - 1.01i)T + (60.3 - 9.09i)T^{2} \) |
| 67 | \( 1 + (-2.96 + 5.13i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + (4.66 + 9.69i)T + (-44.2 + 55.5i)T^{2} \) |
| 73 | \( 1 + (-3.82 - 12.4i)T + (-60.3 + 41.1i)T^{2} \) |
| 79 | \( 1 + (-3.99 - 6.91i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (7.60 + 7.06i)T + (6.20 + 82.7i)T^{2} \) |
| 89 | \( 1 + (-5.96 - 5.53i)T + (6.65 + 88.7i)T^{2} \) |
| 97 | \( 1 + (-11.6 - 6.72i)T + (48.5 + 84.0i)T^{2} \) |
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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
−10.74285595117595526691455502106, −10.23367325728749030203460864364, −9.505248335220964366817278868081, −8.738793115591788153341466743893, −7.71613335250958801159693006241, −6.33238709700768687914929403868, −5.10562776742331600955414423393, −3.80549243509367632003211372501, −2.85578620147535095737336855183, −1.14327541901875752699771862031,
1.19970168846098123115111739315, 2.70067077743719190696858627615, 4.71358112991271357957291628753, 6.09163015895495122508113514490, 6.60430312087512599362519884516, 7.58425077209867553187475767116, 8.408798501208371895679602159317, 9.211087938310044655053090687040, 9.624677881568922779248245313426, 11.63903160448105096471107326145
