| L(s) = 1 | + (−0.819 − 0.573i)2-s + (1.34 − 1.09i)3-s + (0.342 + 0.939i)4-s + (−0.114 + 2.23i)5-s + (−1.72 + 0.129i)6-s + (−4.42 + 1.18i)7-s + (0.258 − 0.965i)8-s + (0.594 − 2.94i)9-s + (1.37 − 1.76i)10-s + (−2.34 + 1.35i)11-s + (1.48 + 0.884i)12-s + (−2.13 + 0.186i)13-s + (4.30 + 1.56i)14-s + (2.29 + 3.11i)15-s + (−0.766 + 0.642i)16-s + (5.48 + 3.83i)17-s + ⋯ |
| L(s) = 1 | + (−0.579 − 0.405i)2-s + (0.774 − 0.633i)3-s + (0.171 + 0.469i)4-s + (−0.0511 + 0.998i)5-s + (−0.705 + 0.0528i)6-s + (−1.67 + 0.448i)7-s + (0.0915 − 0.341i)8-s + (0.198 − 0.980i)9-s + (0.434 − 0.557i)10-s + (−0.706 + 0.407i)11-s + (0.429 + 0.255i)12-s + (−0.591 + 0.0517i)13-s + (1.15 + 0.418i)14-s + (0.592 + 0.805i)15-s + (−0.191 + 0.160i)16-s + (1.32 + 0.930i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0972 - 0.995i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0972 - 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.416518 + 0.459224i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.416518 + 0.459224i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.819 + 0.573i)T \) |
| 3 | \( 1 + (-1.34 + 1.09i)T \) |
| 5 | \( 1 + (0.114 - 2.23i)T \) |
| 19 | \( 1 + (2.70 - 3.41i)T \) |
| good | 7 | \( 1 + (4.42 - 1.18i)T + (6.06 - 3.5i)T^{2} \) |
| 11 | \( 1 + (2.34 - 1.35i)T + (5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (2.13 - 0.186i)T + (12.8 - 2.25i)T^{2} \) |
| 17 | \( 1 + (-5.48 - 3.83i)T + (5.81 + 15.9i)T^{2} \) |
| 23 | \( 1 + (-0.145 - 0.0679i)T + (14.7 + 17.6i)T^{2} \) |
| 29 | \( 1 + (0.197 - 1.11i)T + (-27.2 - 9.91i)T^{2} \) |
| 31 | \( 1 + (5.32 - 9.22i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (4.34 + 4.34i)T + 37iT^{2} \) |
| 41 | \( 1 + (-5.93 - 7.06i)T + (-7.11 + 40.3i)T^{2} \) |
| 43 | \( 1 + (-0.248 + 0.115i)T + (27.6 - 32.9i)T^{2} \) |
| 47 | \( 1 + (1.60 + 2.29i)T + (-16.0 + 44.1i)T^{2} \) |
| 53 | \( 1 + (9.55 + 4.45i)T + (34.0 + 40.6i)T^{2} \) |
| 59 | \( 1 + (-1.11 - 6.33i)T + (-55.4 + 20.1i)T^{2} \) |
| 61 | \( 1 + (-10.8 + 3.93i)T + (46.7 - 39.2i)T^{2} \) |
| 67 | \( 1 + (3.08 - 2.16i)T + (22.9 - 62.9i)T^{2} \) |
| 71 | \( 1 + (3.25 - 8.93i)T + (-54.3 - 45.6i)T^{2} \) |
| 73 | \( 1 + (-1.24 + 14.2i)T + (-71.8 - 12.6i)T^{2} \) |
| 79 | \( 1 + (-0.752 - 0.896i)T + (-13.7 + 77.7i)T^{2} \) |
| 83 | \( 1 + (-7.77 + 2.08i)T + (71.8 - 41.5i)T^{2} \) |
| 89 | \( 1 + (-1.54 - 1.29i)T + (15.4 + 87.6i)T^{2} \) |
| 97 | \( 1 + (-0.808 + 1.15i)T + (-33.1 - 91.1i)T^{2} \) |
| 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
−10.60428405512242726846090160741, −10.03007137223200951104836057498, −9.387662627990253990779428047095, −8.308972334096066057243045859578, −7.44725963151138854024613670542, −6.73490303009016195993125435620, −5.84922496040855290253497557322, −3.59830157034657466886773269788, −3.06735973283284766534483471567, −2.00575074066078807956458322553,
0.35934969775411640110576307936, 2.58058050518432744103532197185, 3.69999105235124227508823137943, 4.93925203065491980617740053389, 5.87244781251928452405019088591, 7.24072800385039842257548941912, 7.894387724150502041847142216932, 8.946397837644550578229196679799, 9.597850261751306946755458845564, 9.994365802793231578896311011274
