| L(s) = 1 | + (−3.19 − 0.185i)2-s + (0.779 − 2.89i)3-s + (6.17 + 0.722i)4-s + (−1.14 − 4.82i)5-s + (−3.02 + 9.10i)6-s + (2.64 + 3.55i)7-s + (−6.99 − 1.23i)8-s + (−7.78 − 4.51i)9-s + (2.75 + 15.5i)10-s + (−7.62 − 7.19i)11-s + (6.90 − 17.3i)12-s + (−18.4 − 9.28i)13-s + (−7.77 − 11.8i)14-s + (−14.8 − 0.447i)15-s + (−2.12 − 0.503i)16-s + (−7.36 + 20.2i)17-s + ⋯ |
| L(s) = 1 | + (−1.59 − 0.0929i)2-s + (0.259 − 0.965i)3-s + (1.54 + 0.180i)4-s + (−0.228 − 0.964i)5-s + (−0.504 + 1.51i)6-s + (0.377 + 0.507i)7-s + (−0.873 − 0.154i)8-s + (−0.864 − 0.501i)9-s + (0.275 + 1.55i)10-s + (−0.693 − 0.653i)11-s + (0.575 − 1.44i)12-s + (−1.42 − 0.714i)13-s + (−0.555 − 0.844i)14-s + (−0.990 − 0.0298i)15-s + (−0.132 − 0.0314i)16-s + (−0.433 + 1.19i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 81 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.903 + 0.427i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 81 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.903 + 0.427i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.0997177 - 0.443924i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0997177 - 0.443924i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-0.779 + 2.89i)T \) |
| good | 2 | \( 1 + (3.19 + 0.185i)T + (3.97 + 0.464i)T^{2} \) |
| 5 | \( 1 + (1.14 + 4.82i)T + (-22.3 + 11.2i)T^{2} \) |
| 7 | \( 1 + (-2.64 - 3.55i)T + (-14.0 + 46.9i)T^{2} \) |
| 11 | \( 1 + (7.62 + 7.19i)T + (7.03 + 120. i)T^{2} \) |
| 13 | \( 1 + (18.4 + 9.28i)T + (100. + 135. i)T^{2} \) |
| 17 | \( 1 + (7.36 - 20.2i)T + (-221. - 185. i)T^{2} \) |
| 19 | \( 1 + (-24.2 + 8.83i)T + (276. - 232. i)T^{2} \) |
| 23 | \( 1 + (4.57 + 3.40i)T + (151. + 506. i)T^{2} \) |
| 29 | \( 1 + (-8.55 + 13.0i)T + (-333. - 772. i)T^{2} \) |
| 31 | \( 1 + (4.91 + 11.4i)T + (-659. + 699. i)T^{2} \) |
| 37 | \( 1 + (2.09 - 1.75i)T + (237. - 1.34e3i)T^{2} \) |
| 41 | \( 1 + (-46.4 + 2.70i)T + (1.66e3 - 195. i)T^{2} \) |
| 43 | \( 1 + (17.6 + 59.0i)T + (-1.54e3 + 1.01e3i)T^{2} \) |
| 47 | \( 1 + (20.6 + 8.89i)T + (1.51e3 + 1.60e3i)T^{2} \) |
| 53 | \( 1 + (-80.0 + 46.2i)T + (1.40e3 - 2.43e3i)T^{2} \) |
| 59 | \( 1 + (18.0 - 17.0i)T + (202. - 3.47e3i)T^{2} \) |
| 61 | \( 1 + (-114. + 13.3i)T + (3.62e3 - 858. i)T^{2} \) |
| 67 | \( 1 + (90.6 - 59.6i)T + (1.77e3 - 4.12e3i)T^{2} \) |
| 71 | \( 1 + (-52.3 + 9.23i)T + (4.73e3 - 1.72e3i)T^{2} \) |
| 73 | \( 1 + (13.3 - 75.4i)T + (-5.00e3 - 1.82e3i)T^{2} \) |
| 79 | \( 1 + (-1.48 + 25.5i)T + (-6.19e3 - 724. i)T^{2} \) |
| 83 | \( 1 + (-39.2 - 2.28i)T + (6.84e3 + 799. i)T^{2} \) |
| 89 | \( 1 + (116. + 20.5i)T + (7.44e3 + 2.70e3i)T^{2} \) |
| 97 | \( 1 + (-128. - 30.5i)T + (8.40e3 + 4.22e3i)T^{2} \) |
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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
−13.38570645498146795432033388371, −12.35481115676641308965951222624, −11.44653358674104776306991047267, −10.04669795431175796550479341667, −8.747262968720047520282565424986, −8.213879269467770568133577338223, −7.25977426051314302982637133888, −5.42805398839893878381046928992, −2.35948484359058857339612151561, −0.58996860290661982384561206053,
2.64909087566657431672581702336, 4.77390812845307683604983075344, 7.14030908533821028272116260355, 7.72192844860552868762378993528, 9.301024745968646228449788042569, 9.962275536593035223048469042030, 10.83017825947378063264426289177, 11.72831596046026989037267646594, 13.98476722585306105474488458791, 14.81839902021173749721994642065
