L(s) = 1 | − 6.80·5-s + 9.74·7-s − 20.3·11-s + 24.2i·13-s + 21.2·17-s + (15.2 + 11.3i)19-s + 20.7·23-s + 21.3·25-s + 21.9i·29-s − 40.0i·31-s − 66.3·35-s − 44.5i·37-s + 17.7i·41-s − 4.69·43-s − 1.15·47-s + ⋯ |
L(s) = 1 | − 1.36·5-s + 1.39·7-s − 1.85·11-s + 1.86i·13-s + 1.24·17-s + (0.800 + 0.598i)19-s + 0.900·23-s + 0.853·25-s + 0.755i·29-s − 1.29i·31-s − 1.89·35-s − 1.20i·37-s + 0.432i·41-s − 0.109·43-s − 0.0244·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.800 - 0.598i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.800 - 0.598i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(0.8733915362\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.8733915362\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 19 | \( 1 + (-15.2 - 11.3i)T \) |
good | 5 | \( 1 + 6.80T + 25T^{2} \) |
| 7 | \( 1 - 9.74T + 49T^{2} \) |
| 11 | \( 1 + 20.3T + 121T^{2} \) |
| 13 | \( 1 - 24.2iT - 169T^{2} \) |
| 17 | \( 1 - 21.2T + 289T^{2} \) |
| 23 | \( 1 - 20.7T + 529T^{2} \) |
| 29 | \( 1 - 21.9iT - 841T^{2} \) |
| 31 | \( 1 + 40.0iT - 961T^{2} \) |
| 37 | \( 1 + 44.5iT - 1.36e3T^{2} \) |
| 41 | \( 1 - 17.7iT - 1.68e3T^{2} \) |
| 43 | \( 1 + 4.69T + 1.84e3T^{2} \) |
| 47 | \( 1 + 1.15T + 2.20e3T^{2} \) |
| 53 | \( 1 + 75.9iT - 2.80e3T^{2} \) |
| 59 | \( 1 - 0.973iT - 3.48e3T^{2} \) |
| 61 | \( 1 + 77.0T + 3.72e3T^{2} \) |
| 67 | \( 1 - 33.2iT - 4.48e3T^{2} \) |
| 71 | \( 1 - 91.7iT - 5.04e3T^{2} \) |
| 73 | \( 1 + 97.9T + 5.32e3T^{2} \) |
| 79 | \( 1 - 43.0iT - 6.24e3T^{2} \) |
| 83 | \( 1 - 55.7T + 6.88e3T^{2} \) |
| 89 | \( 1 - 74.9iT - 7.92e3T^{2} \) |
| 97 | \( 1 - 55.2iT - 9.40e3T^{2} \) |
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
−8.728243065748875200835614384397, −7.928690429201775516754432335002, −7.68529193358491286685948583505, −7.02180034078121593170930310522, −5.60649708477216985371817037937, −4.99268643585350883071211738520, −4.27963722821050097893539570442, −3.43157662421831674385608944453, −2.29473742710039702617050824702, −1.17991540694079081491584833499,
0.23700980254556763050100147727, 1.21268494102661488022362341427, 2.90800009764284977605104246317, 3.21695225664950769407270149132, 4.67391513931177492211086084627, 5.07086968865218500537467169358, 5.73349137383493712830942389234, 7.36300391965827230752028934708, 7.73912536952151852118219727408, 8.024542610229648072500918732974