L(s) = 1 | + (−0.5 + 0.866i)2-s + (1.5 − 0.866i)3-s + (0.500 + 0.866i)4-s + 5-s + 1.73i·6-s − 3·8-s + (1.5 − 2.59i)9-s + (−0.5 + 0.866i)10-s + 5·11-s + (1.5 + 0.866i)12-s + (−2.5 + 4.33i)13-s + (1.5 − 0.866i)15-s + (0.500 − 0.866i)16-s + (1.5 − 2.59i)17-s + (1.5 + 2.59i)18-s + (0.5 + 0.866i)19-s + ⋯ |
L(s) = 1 | + (−0.353 + 0.612i)2-s + (0.866 − 0.499i)3-s + (0.250 + 0.433i)4-s + 0.447·5-s + 0.707i·6-s − 1.06·8-s + (0.5 − 0.866i)9-s + (−0.158 + 0.273i)10-s + 1.50·11-s + (0.433 + 0.249i)12-s + (−0.693 + 1.20i)13-s + (0.387 − 0.223i)15-s + (0.125 − 0.216i)16-s + (0.363 − 0.630i)17-s + (0.353 + 0.612i)18-s + (0.114 + 0.198i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.678 - 0.734i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.678 - 0.734i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.64937 + 0.721660i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.64937 + 0.721660i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (-1.5 + 0.866i)T \) |
| 7 | \( 1 \) |
good | 2 | \( 1 + (0.5 - 0.866i)T + (-1 - 1.73i)T^{2} \) |
| 5 | \( 1 - T + 5T^{2} \) |
| 11 | \( 1 - 5T + 11T^{2} \) |
| 13 | \( 1 + (2.5 - 4.33i)T + (-6.5 - 11.2i)T^{2} \) |
| 17 | \( 1 + (-1.5 + 2.59i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (-0.5 - 0.866i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 - 3T + 23T^{2} \) |
| 29 | \( 1 + (-0.5 - 0.866i)T + (-14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (1.5 + 2.59i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (2.5 - 4.33i)T + (-20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (-0.5 - 0.866i)T + (-21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (-4.5 + 7.79i)T + (-26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (7 - 12.1i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (2 + 3.46i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + 12T + 71T^{2} \) |
| 73 | \( 1 + (-1.5 + 2.59i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (4 - 6.92i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (4.5 + 7.79i)T + (-41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 + (6.5 + 11.2i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (4.5 + 7.79i)T + (-48.5 + 84.0i)T^{2} \) |
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
−11.67270127515268801840036541869, −9.813220415498522251491742169714, −9.207429231171912215799198859884, −8.584233361275081863493018719485, −7.36975048322615706487256264248, −6.91988566287984603458181768886, −6.01572280651773631726681495099, −4.22808360698426185127864388297, −3.05696271788775605588001218213, −1.74641034693958679014655227369,
1.45886971208607556344196644717, 2.69016880952267537302195258300, 3.75520227418099950711137063037, 5.20137328974966819605129078249, 6.26463411179910811009541442903, 7.47292159610009580697029651338, 8.666517176668105822070017843953, 9.385078647897478007497369838284, 10.05889829499358720750331639581, 10.68464167078627835701625326494