| L(s) = 1 | + 2.61i·2-s − 2.86·4-s + 0.807i·5-s + 5.91·7-s + 2.97i·8-s − 2.11·10-s − 18.9i·11-s + 19.7·13-s + 15.5i·14-s − 19.2·16-s + 4.67i·17-s + 9.92·19-s − 2.31i·20-s + 49.7·22-s − 12.7i·23-s + ⋯ |
| L(s) = 1 | + 1.30i·2-s − 0.715·4-s + 0.161i·5-s + 0.845·7-s + 0.372i·8-s − 0.211·10-s − 1.72i·11-s + 1.51·13-s + 1.10i·14-s − 1.20·16-s + 0.274i·17-s + 0.522·19-s − 0.115i·20-s + 2.25·22-s − 0.555i·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 729 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -i\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 729 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & -i\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(2.304993662\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.304993662\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| good | 2 | \( 1 - 2.61iT - 4T^{2} \) |
| 5 | \( 1 - 0.807iT - 25T^{2} \) |
| 7 | \( 1 - 5.91T + 49T^{2} \) |
| 11 | \( 1 + 18.9iT - 121T^{2} \) |
| 13 | \( 1 - 19.7T + 169T^{2} \) |
| 17 | \( 1 - 4.67iT - 289T^{2} \) |
| 19 | \( 1 - 9.92T + 361T^{2} \) |
| 23 | \( 1 + 12.7iT - 529T^{2} \) |
| 29 | \( 1 + 6.03iT - 841T^{2} \) |
| 31 | \( 1 - 36.6T + 961T^{2} \) |
| 37 | \( 1 + 18.8T + 1.36e3T^{2} \) |
| 41 | \( 1 + 22.0iT - 1.68e3T^{2} \) |
| 43 | \( 1 - 34.7T + 1.84e3T^{2} \) |
| 47 | \( 1 - 28.9iT - 2.20e3T^{2} \) |
| 53 | \( 1 - 19.9iT - 2.80e3T^{2} \) |
| 59 | \( 1 - 59.8iT - 3.48e3T^{2} \) |
| 61 | \( 1 + 14.8T + 3.72e3T^{2} \) |
| 67 | \( 1 - 58.3T + 4.48e3T^{2} \) |
| 71 | \( 1 - 83.6iT - 5.04e3T^{2} \) |
| 73 | \( 1 + 114.T + 5.32e3T^{2} \) |
| 79 | \( 1 - 28.5T + 6.24e3T^{2} \) |
| 83 | \( 1 + 23.3iT - 6.88e3T^{2} \) |
| 89 | \( 1 - 140. iT - 7.92e3T^{2} \) |
| 97 | \( 1 - 66.0T + 9.40e3T^{2} \) |
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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
−10.65344926309182953192148783659, −9.000835019333395175733257849944, −8.434374753598654924474373206314, −7.945494758217491560501990453351, −6.77023949474183503979873941005, −6.03910510335037628117979112121, −5.39038794587028176247253965620, −4.19770360326663418963646753487, −2.90139648982296102330485507628, −1.08847248668805792028241061479,
1.14046183344873734430860926867, 1.93994637160376387868650159157, 3.20964662216982804096605568432, 4.30434696317560716059485220279, 5.04564989538372870545215259208, 6.50796156459858813567359773275, 7.44116375498038278913661916465, 8.499929533512307937803268254981, 9.393095794537814696699557909529, 10.14334417796438643472428608746
