L(s) = 1 | + (−0.5 − 0.866i)2-s + (−0.499 + 0.866i)4-s − 1.37·5-s + 0.999·8-s + (0.686 + 1.18i)10-s − 4.37·11-s + (1 + 1.73i)13-s + (−0.5 − 0.866i)16-s + (2.18 + 3.78i)17-s + (2.5 − 4.33i)19-s + (0.686 − 1.18i)20-s + (2.18 + 3.78i)22-s + 7.37·23-s − 3.11·25-s + (0.999 − 1.73i)26-s + ⋯ |
L(s) = 1 | + (−0.353 − 0.612i)2-s + (−0.249 + 0.433i)4-s − 0.613·5-s + 0.353·8-s + (0.216 + 0.375i)10-s − 1.31·11-s + (0.277 + 0.480i)13-s + (−0.125 − 0.216i)16-s + (0.530 + 0.918i)17-s + (0.573 − 0.993i)19-s + (0.153 − 0.265i)20-s + (0.466 + 0.807i)22-s + 1.53·23-s − 0.623·25-s + (0.196 − 0.339i)26-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2646 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.988 - 0.151i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2646 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.988 - 0.151i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.2205679898\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.2205679898\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.5 + 0.866i)T \) |
| 3 | \( 1 \) |
| 7 | \( 1 \) |
good | 5 | \( 1 + 1.37T + 5T^{2} \) |
| 11 | \( 1 + 4.37T + 11T^{2} \) |
| 13 | \( 1 + (-1 - 1.73i)T + (-6.5 + 11.2i)T^{2} \) |
| 17 | \( 1 + (-2.18 - 3.78i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (-2.5 + 4.33i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 - 7.37T + 23T^{2} \) |
| 29 | \( 1 + (-1.37 + 2.37i)T + (-14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + (-1 + 1.73i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (1 - 1.73i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (5.18 + 8.98i)T + (-20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (4.55 - 7.89i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (-1.37 - 2.37i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (3.55 - 6.16i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (7.05 + 12.2i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (7.55 - 13.0i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + 10.1T + 71T^{2} \) |
| 73 | \( 1 + (2.55 + 4.43i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (6.05 + 10.4i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (-2.74 + 4.75i)T + (-41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 + (1.62 - 2.81i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (-4.55 + 7.89i)T + (-48.5 - 84.0i)T^{2} \) |
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
−8.566244284610334413542794391053, −7.72442662880701774210480160200, −7.27944090865775516762036080413, −6.15744491760836038406482169502, −5.13864956846164832792509945448, −4.42416136297729393736808409442, −3.38250798146879301115156417147, −2.70919360551981538874235039769, −1.45549595698793833188549772959, −0.090289593067936402860951400937,
1.24084607795543531909602264880, 2.82150446605565505428618394921, 3.56697090247335501410285006059, 4.91104502472246924977448718336, 5.28276044631606838093298756513, 6.23126885540661381447017695455, 7.29828602054554547070065834327, 7.64657677336109995884671063366, 8.349680819714822198987508077116, 9.073684140267707047390624338360