| L(s) = 1 | − 1.04i·2-s + 14.9·4-s − 11.1i·5-s + 72.9·7-s − 32.2i·8-s − 11.6·10-s + 141. i·11-s − 6.18·13-s − 76.2i·14-s + 204.·16-s + 452. i·17-s + 621.·19-s − 166. i·20-s + 147.·22-s − 32.3i·23-s + ⋯ |
| L(s) = 1 | − 0.261i·2-s + 0.931·4-s − 0.447i·5-s + 1.48·7-s − 0.504i·8-s − 0.116·10-s + 1.17i·11-s − 0.0365·13-s − 0.389i·14-s + 0.800·16-s + 1.56i·17-s + 1.72·19-s − 0.416i·20-s + 0.305·22-s − 0.0611i·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 405 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 405 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & \, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(3.295615509\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.295615509\) |
| \(L(3)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 + 11.1iT \) |
| good | 2 | \( 1 + 1.04iT - 16T^{2} \) |
| 7 | \( 1 - 72.9T + 2.40e3T^{2} \) |
| 11 | \( 1 - 141. iT - 1.46e4T^{2} \) |
| 13 | \( 1 + 6.18T + 2.85e4T^{2} \) |
| 17 | \( 1 - 452. iT - 8.35e4T^{2} \) |
| 19 | \( 1 - 621.T + 1.30e5T^{2} \) |
| 23 | \( 1 + 32.3iT - 2.79e5T^{2} \) |
| 29 | \( 1 - 1.00e3iT - 7.07e5T^{2} \) |
| 31 | \( 1 + 1.36e3T + 9.23e5T^{2} \) |
| 37 | \( 1 + 711.T + 1.87e6T^{2} \) |
| 41 | \( 1 + 1.05e3iT - 2.82e6T^{2} \) |
| 43 | \( 1 + 237.T + 3.41e6T^{2} \) |
| 47 | \( 1 - 2.50e3iT - 4.87e6T^{2} \) |
| 53 | \( 1 + 1.12e3iT - 7.89e6T^{2} \) |
| 59 | \( 1 + 1.84e3iT - 1.21e7T^{2} \) |
| 61 | \( 1 - 5.53e3T + 1.38e7T^{2} \) |
| 67 | \( 1 + 808.T + 2.01e7T^{2} \) |
| 71 | \( 1 + 1.75e3iT - 2.54e7T^{2} \) |
| 73 | \( 1 + 1.90e3T + 2.83e7T^{2} \) |
| 79 | \( 1 + 3.63e3T + 3.89e7T^{2} \) |
| 83 | \( 1 + 9.77e3iT - 4.74e7T^{2} \) |
| 89 | \( 1 + 2.45e3iT - 6.27e7T^{2} \) |
| 97 | \( 1 - 1.35e4T + 8.85e7T^{2} \) |
| 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
−10.77553579496248105294234475790, −9.917957162111492595455784946632, −8.740892744225027521038399079659, −7.71275630541064469028786922610, −7.13876747853779834799656426688, −5.70076117522492424652815048285, −4.82524953971956814842148383815, −3.57181931963427794414194335758, −1.92977754691825382037857522872, −1.38558197591081398702994068317,
1.00299114789955362472840473008, 2.32779809197070244972379982763, 3.41445253219053787477732807471, 5.09867015790081160046872639884, 5.76365887552603888973205393863, 7.11334043085420235265815698422, 7.64980785534690631115731823942, 8.570029104669726120891892819064, 9.809553824453405698264975617181, 10.96242396450610524631340090814
