L(s) = 1 | + (−0.400 + 0.694i)2-s + (0.678 + 1.17i)4-s + (−1.37 − 2.38i)5-s + (−1.18 + 2.05i)7-s − 2.69·8-s + 2.20·10-s + (−0.125 + 0.216i)11-s + (−1.30 − 2.26i)13-s + (−0.952 − 1.64i)14-s + (−0.279 + 0.483i)16-s + 0.293·17-s − 2.78·19-s + (1.86 − 3.23i)20-s + (−0.100 − 0.173i)22-s + (−3.34 − 5.79i)23-s + ⋯ |
L(s) = 1 | + (−0.283 + 0.490i)2-s + (0.339 + 0.587i)4-s + (−0.614 − 1.06i)5-s + (−0.449 + 0.778i)7-s − 0.951·8-s + 0.696·10-s + (−0.0377 + 0.0653i)11-s + (−0.362 − 0.627i)13-s + (−0.254 − 0.440i)14-s + (−0.0697 + 0.120i)16-s + 0.0711·17-s − 0.638·19-s + (0.417 − 0.722i)20-s + (−0.0213 − 0.0370i)22-s + (−0.697 − 1.20i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 729 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.500 + 0.866i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 729 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.500 + 0.866i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.0880435 - 0.152495i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.0880435 - 0.152495i\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + (0.400 - 0.694i)T + (-1 - 1.73i)T^{2} \) |
| 5 | \( 1 + (1.37 + 2.38i)T + (-2.5 + 4.33i)T^{2} \) |
| 7 | \( 1 + (1.18 - 2.05i)T + (-3.5 - 6.06i)T^{2} \) |
| 11 | \( 1 + (0.125 - 0.216i)T + (-5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (1.30 + 2.26i)T + (-6.5 + 11.2i)T^{2} \) |
| 17 | \( 1 - 0.293T + 17T^{2} \) |
| 19 | \( 1 + 2.78T + 19T^{2} \) |
| 23 | \( 1 + (3.34 + 5.79i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (0.177 - 0.307i)T + (-14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + (1.38 + 2.39i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + 6.99T + 37T^{2} \) |
| 41 | \( 1 + (4.85 + 8.41i)T + (-20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (0.130 - 0.225i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (5.71 - 9.89i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + 5.43T + 53T^{2} \) |
| 59 | \( 1 + (-2.98 - 5.17i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (-5.92 + 10.2i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (0.905 + 1.56i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 - 0.370T + 71T^{2} \) |
| 73 | \( 1 - 5.02T + 73T^{2} \) |
| 79 | \( 1 + (0.401 - 0.695i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (1.37 - 2.38i)T + (-41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 + 10.4T + 89T^{2} \) |
| 97 | \( 1 + (-7.41 + 12.8i)T + (-48.5 - 84.0i)T^{2} \) |
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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
−9.892962440413769166670030666594, −8.867828171118414686086753262619, −8.433573062152157329395082050505, −7.70116338625613734149993779369, −6.64963554133890496384072568700, −5.75562497286351360253912970383, −4.67430574352447575976781576868, −3.52515283886509631283419060348, −2.32528668484946959788396558474, −0.092180150973352895578078890732,
1.77908412772659878611026265633, 3.09045068019826628686486209710, 3.89833683024580891661646719024, 5.35834665072322745943830457058, 6.67125493816066222460853759762, 6.91229779248513226956545761802, 8.042420648044037013872479326124, 9.255659171326861038644332801584, 10.14380268589553958292663657669, 10.51291922408787358274431043496