| L(s) = 1 | − 3i·2-s − 5·4-s + 3i·5-s + 5·7-s + 3i·8-s + 9·10-s + 15i·11-s − 10·13-s − 15i·14-s − 11·16-s − 18i·17-s − 16·19-s − 15i·20-s + 45·22-s + 12i·23-s + ⋯ |
| L(s) = 1 | − 1.5i·2-s − 1.25·4-s + 0.600i·5-s + 0.714·7-s + 0.375i·8-s + 0.900·10-s + 1.36i·11-s − 0.769·13-s − 1.07i·14-s − 0.687·16-s − 1.05i·17-s − 0.842·19-s − 0.750i·20-s + 2.04·22-s + 0.521i·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 27 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & i\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 27 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & i\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.670334 - 0.670334i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.670334 - 0.670334i\) |
| \(L(2)\) |
|
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 | \( 1 + 3iT - 4T^{2} \) |
| 5 | \( 1 - 3iT - 25T^{2} \) |
| 7 | \( 1 - 5T + 49T^{2} \) |
| 11 | \( 1 - 15iT - 121T^{2} \) |
| 13 | \( 1 + 10T + 169T^{2} \) |
| 17 | \( 1 + 18iT - 289T^{2} \) |
| 19 | \( 1 + 16T + 361T^{2} \) |
| 23 | \( 1 - 12iT - 529T^{2} \) |
| 29 | \( 1 + 30iT - 841T^{2} \) |
| 31 | \( 1 + T + 961T^{2} \) |
| 37 | \( 1 - 20T + 1.36e3T^{2} \) |
| 41 | \( 1 + 60iT - 1.68e3T^{2} \) |
| 43 | \( 1 - 50T + 1.84e3T^{2} \) |
| 47 | \( 1 - 6iT - 2.20e3T^{2} \) |
| 53 | \( 1 - 27iT - 2.80e3T^{2} \) |
| 59 | \( 1 - 30iT - 3.48e3T^{2} \) |
| 61 | \( 1 + 76T + 3.72e3T^{2} \) |
| 67 | \( 1 + 10T + 4.48e3T^{2} \) |
| 71 | \( 1 - 90iT - 5.04e3T^{2} \) |
| 73 | \( 1 - 65T + 5.32e3T^{2} \) |
| 79 | \( 1 - 14T + 6.24e3T^{2} \) |
| 83 | \( 1 + 3iT - 6.88e3T^{2} \) |
| 89 | \( 1 + 90iT - 7.92e3T^{2} \) |
| 97 | \( 1 + 85T + 9.40e3T^{2} \) |
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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
−17.36808523137500384684965942861, −15.32702670319768498329699627984, −14.14876383614042419244364506857, −12.65414210033438109237929814485, −11.65390726870685934002347607507, −10.52950361117322614987594461701, −9.420708282430980154915816688071, −7.28542109883854820063536672271, −4.53354833528689371647994175977, −2.36734171042264431836895026524,
4.83051333614660063586439682390, 6.24922237424642260720064150063, 7.981613995047342787015509059394, 8.829029835700150106568585551216, 10.99282694792126830280638291561, 12.78777173527289554999790494628, 14.21316958405189901201418238593, 15.02205089555784189285554512850, 16.44271949144973078182113659631, 16.95867641883435699347127869064
