| L(s) = 1 | + 14.1i·2-s − 134.·4-s − 137. i·5-s − 514.·7-s − 1.00e3i·8-s + 1.93e3·10-s + 1.21e3i·11-s − 170.·13-s − 7.26e3i·14-s + 5.47e3·16-s − 1.57e3i·17-s − 1.12e4·19-s + 1.85e4i·20-s − 1.71e4·22-s + 1.05e4i·23-s + ⋯ |
| L(s) = 1 | + 1.76i·2-s − 2.10·4-s − 1.09i·5-s − 1.50·7-s − 1.95i·8-s + 1.93·10-s + 0.915i·11-s − 0.0775·13-s − 2.64i·14-s + 1.33·16-s − 0.320i·17-s − 1.64·19-s + 2.31i·20-s − 1.61·22-s + 0.868i·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 27 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & i\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 27 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & i\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{7}{2})\) |
\(\approx\) |
\(0.0902390 - 0.0902390i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0902390 - 0.0902390i\) |
| \(L(4)\) |
|
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 | \( 1 - 14.1iT - 64T^{2} \) |
| 5 | \( 1 + 137. iT - 1.56e4T^{2} \) |
| 7 | \( 1 + 514.T + 1.17e5T^{2} \) |
| 11 | \( 1 - 1.21e3iT - 1.77e6T^{2} \) |
| 13 | \( 1 + 170.T + 4.82e6T^{2} \) |
| 17 | \( 1 + 1.57e3iT - 2.41e7T^{2} \) |
| 19 | \( 1 + 1.12e4T + 4.70e7T^{2} \) |
| 23 | \( 1 - 1.05e4iT - 1.48e8T^{2} \) |
| 29 | \( 1 + 8.66e3iT - 5.94e8T^{2} \) |
| 31 | \( 1 + 2.62e4T + 8.87e8T^{2} \) |
| 37 | \( 1 - 2.18e4T + 2.56e9T^{2} \) |
| 41 | \( 1 - 3.81e4iT - 4.75e9T^{2} \) |
| 43 | \( 1 - 2.38e4T + 6.32e9T^{2} \) |
| 47 | \( 1 + 2.66e4iT - 1.07e10T^{2} \) |
| 53 | \( 1 - 2.13e5iT - 2.21e10T^{2} \) |
| 59 | \( 1 + 1.63e5iT - 4.21e10T^{2} \) |
| 61 | \( 1 - 1.11e5T + 5.15e10T^{2} \) |
| 67 | \( 1 + 3.81e5T + 9.04e10T^{2} \) |
| 71 | \( 1 - 6.75e5iT - 1.28e11T^{2} \) |
| 73 | \( 1 + 7.62e5T + 1.51e11T^{2} \) |
| 79 | \( 1 - 5.15e5T + 2.43e11T^{2} \) |
| 83 | \( 1 + 9.03e5iT - 3.26e11T^{2} \) |
| 89 | \( 1 + 6.44e5iT - 4.96e11T^{2} \) |
| 97 | \( 1 + 6.16e5T + 8.32e11T^{2} \) |
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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
−16.67062491574330555249036819393, −15.90607050091366008886217608971, −14.90006550890958373833857342774, −13.28758399739109253286011482215, −12.62573745941034360210177811473, −9.712757088698401587196219388996, −8.774490473079452670235927196588, −7.22269592426998206064569776166, −5.94911344492520936529687517733, −4.42432489056320558551390835928,
0.06990772488473806550174332336, 2.61148876668330449667207642132, 3.71439030626939571547904683655, 6.42938884155899717089417501161, 8.924972782933606673051086079242, 10.29759850575312752134050842379, 10.94380486664315432848797359820, 12.47181958018302089082748430914, 13.35217254499857176157784018278, 14.70602077666182556688517006087
