| L(s) = 1 | + 0.909·5-s + 7.05i·7-s + 8.04i·11-s − 6.71·13-s − 26.3·17-s − 20.5i·19-s − 25.2i·23-s − 24.1·25-s − 30.3·29-s − 0.138i·31-s + 6.41i·35-s + 69.7·37-s + 58.7·41-s − 2.83i·43-s − 81.7i·47-s + ⋯ |
| L(s) = 1 | + 0.181·5-s + 1.00i·7-s + 0.731i·11-s − 0.516·13-s − 1.54·17-s − 1.08i·19-s − 1.09i·23-s − 0.966·25-s − 1.04·29-s − 0.00447i·31-s + 0.183i·35-s + 1.88·37-s + 1.43·41-s − 0.0660i·43-s − 1.73i·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1296 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.5 + 0.866i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1296 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.5 + 0.866i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.5481750772\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5481750772\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 - 0.909T + 25T^{2} \) |
| 7 | \( 1 - 7.05iT - 49T^{2} \) |
| 11 | \( 1 - 8.04iT - 121T^{2} \) |
| 13 | \( 1 + 6.71T + 169T^{2} \) |
| 17 | \( 1 + 26.3T + 289T^{2} \) |
| 19 | \( 1 + 20.5iT - 361T^{2} \) |
| 23 | \( 1 + 25.2iT - 529T^{2} \) |
| 29 | \( 1 + 30.3T + 841T^{2} \) |
| 31 | \( 1 + 0.138iT - 961T^{2} \) |
| 37 | \( 1 - 69.7T + 1.36e3T^{2} \) |
| 41 | \( 1 - 58.7T + 1.68e3T^{2} \) |
| 43 | \( 1 + 2.83iT - 1.84e3T^{2} \) |
| 47 | \( 1 + 81.7iT - 2.20e3T^{2} \) |
| 53 | \( 1 + 30.0T + 2.80e3T^{2} \) |
| 59 | \( 1 + 89.0iT - 3.48e3T^{2} \) |
| 61 | \( 1 + 48.1T + 3.72e3T^{2} \) |
| 67 | \( 1 - 50.9iT - 4.48e3T^{2} \) |
| 71 | \( 1 + 68.4iT - 5.04e3T^{2} \) |
| 73 | \( 1 + 22.1T + 5.32e3T^{2} \) |
| 79 | \( 1 - 39.7iT - 6.24e3T^{2} \) |
| 83 | \( 1 - 26.6iT - 6.88e3T^{2} \) |
| 89 | \( 1 + 25.7T + 7.92e3T^{2} \) |
| 97 | \( 1 - 104.T + 9.40e3T^{2} \) |
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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
−9.268863690994102115109599299203, −8.526121178069747268840650860447, −7.53034491000742468023630264452, −6.68213458727548245332664458438, −5.89210466302655586860159430256, −4.89203267477660413109997661425, −4.17822749857053386474371700480, −2.57785375829036520533825808538, −2.10661516195568609158312290712, −0.15537484127315927606603129772,
1.27357240422336219416312808818, 2.54671198330668283032697719335, 3.81737386698756604943103513004, 4.42888776411686285904457969889, 5.72374234796173539861485097108, 6.32499868355177504141750982715, 7.54104896816496085257688449090, 7.82057882810790957722668346435, 9.136480883754454145730684447543, 9.613033454649691717397901382083
