| L(s) = 1 | + 3.15i·2-s − 5.92·4-s − 0.888i·5-s − 10.5·7-s − 6.06i·8-s + 2.79·10-s − 9.89i·11-s + 12.5·13-s − 33.2i·14-s − 4.59·16-s − 4.12i·17-s − 10.5·19-s + 5.26i·20-s + 31.1·22-s − 18.6i·23-s + ⋯ |
| L(s) = 1 | + 1.57i·2-s − 1.48·4-s − 0.177i·5-s − 1.50·7-s − 0.758i·8-s + 0.279·10-s − 0.899i·11-s + 0.964·13-s − 2.37i·14-s − 0.287·16-s − 0.242i·17-s − 0.552·19-s + 0.263i·20-s + 1.41·22-s − 0.809i·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 459 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 459 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & \, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.8158261465\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8158261465\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 17 | \( 1 + 4.12iT \) |
| good | 2 | \( 1 - 3.15iT - 4T^{2} \) |
| 5 | \( 1 + 0.888iT - 25T^{2} \) |
| 7 | \( 1 + 10.5T + 49T^{2} \) |
| 11 | \( 1 + 9.89iT - 121T^{2} \) |
| 13 | \( 1 - 12.5T + 169T^{2} \) |
| 19 | \( 1 + 10.5T + 361T^{2} \) |
| 23 | \( 1 + 18.6iT - 529T^{2} \) |
| 29 | \( 1 + 29.4iT - 841T^{2} \) |
| 31 | \( 1 - 48.9T + 961T^{2} \) |
| 37 | \( 1 + 10.0T + 1.36e3T^{2} \) |
| 41 | \( 1 + 20.6iT - 1.68e3T^{2} \) |
| 43 | \( 1 + 61.9T + 1.84e3T^{2} \) |
| 47 | \( 1 + 2.82iT - 2.20e3T^{2} \) |
| 53 | \( 1 + 59.1iT - 2.80e3T^{2} \) |
| 59 | \( 1 + 38.6iT - 3.48e3T^{2} \) |
| 61 | \( 1 - 14.2T + 3.72e3T^{2} \) |
| 67 | \( 1 + 126.T + 4.48e3T^{2} \) |
| 71 | \( 1 - 25.5iT - 5.04e3T^{2} \) |
| 73 | \( 1 - 16.3T + 5.32e3T^{2} \) |
| 79 | \( 1 + 57.9T + 6.24e3T^{2} \) |
| 83 | \( 1 - 123. iT - 6.88e3T^{2} \) |
| 89 | \( 1 + 101. iT - 7.92e3T^{2} \) |
| 97 | \( 1 + 58.6T + 9.40e3T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.64312151946048587485298249884, −9.642361381229028542465122863872, −8.664788166989331872116884139119, −8.217720564260776235921079742293, −6.73172963053980217749122400433, −6.44543652943486200408587757776, −5.54467632512746753278357489875, −4.25991548177887170158467217046, −3.00141944200620196682159074302, −0.36035556942270560544188090356,
1.35061351958632488808839696923, 2.80478720115363659626580871289, 3.53768611839849516175284697714, 4.62575890397485868145044647987, 6.21129335607539046082072557864, 7.02501527333065461536609214362, 8.612494706753617901850579969388, 9.404969529638381738345453514272, 10.19070272739068673057461685821, 10.68956452380914152077277348759
