L(s) = 1 | + (−0.5 − 0.866i)2-s + (−0.499 + 0.866i)4-s + (−0.571 + 2.16i)5-s + (0.603 − 1.04i)7-s + 0.999·8-s + (2.15 − 0.585i)10-s + (−4.46 + 2.57i)11-s + (2.24 + 2.82i)13-s − 1.20·14-s + (−0.5 − 0.866i)16-s + (−4.10 − 2.36i)17-s + (−1.84 − 1.06i)19-s + (−1.58 − 1.57i)20-s + (4.46 + 2.57i)22-s + (1.88 − 1.08i)23-s + ⋯ |
L(s) = 1 | + (−0.353 − 0.612i)2-s + (−0.249 + 0.433i)4-s + (−0.255 + 0.966i)5-s + (0.227 − 0.394i)7-s + 0.353·8-s + (0.682 − 0.185i)10-s + (−1.34 + 0.776i)11-s + (0.622 + 0.782i)13-s − 0.322·14-s + (−0.125 − 0.216i)16-s + (−0.994 − 0.574i)17-s + (−0.423 − 0.244i)19-s + (−0.354 − 0.352i)20-s + (0.951 + 0.549i)22-s + (0.392 − 0.226i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1170 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.992 - 0.122i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1170 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.992 - 0.122i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.002882025889\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.002882025889\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 5 | \( 1 + (0.571 - 2.16i)T \) |
| 13 | \( 1 + (-2.24 - 2.82i)T \) |
good | 7 | \( 1 + (-0.603 + 1.04i)T + (-3.5 - 6.06i)T^{2} \) |
| 11 | \( 1 + (4.46 - 2.57i)T + (5.5 - 9.52i)T^{2} \) |
| 17 | \( 1 + (4.10 + 2.36i)T + (8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (1.84 + 1.06i)T + (9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (-1.88 + 1.08i)T + (11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (2.38 + 4.12i)T + (-14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + 5.91iT - 31T^{2} \) |
| 37 | \( 1 + (2.20 + 3.81i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (4.10 - 2.36i)T + (20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (-1.70 - 0.986i)T + (21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + 0.852T + 47T^{2} \) |
| 53 | \( 1 + 4.48iT - 53T^{2} \) |
| 59 | \( 1 + (1.68 + 0.970i)T + (29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (1.53 - 2.66i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (7.02 + 12.1i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (-0.298 - 0.172i)T + (35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + 15.7T + 73T^{2} \) |
| 79 | \( 1 + 13.5T + 79T^{2} \) |
| 83 | \( 1 - 10.2T + 83T^{2} \) |
| 89 | \( 1 + (14.1 - 8.19i)T + (44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (4.24 - 7.34i)T + (-48.5 - 84.0i)T^{2} \) |
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
−9.532035885997212358308082744216, −8.593596491771410750862929993101, −7.63351403793453755542243773975, −7.12943786476857914378902355879, −6.13106945256818775082057556838, −4.73211251090412357925615893556, −4.02370654188288838484395124423, −2.77594849123679850352361154004, −2.01346154133665139958262465012, −0.00136425042594675127161682147,
1.52773049885605353677150934004, 3.09294379047329820791689446443, 4.38286681403226424581741556024, 5.38317970082143946178569117736, 5.76695044228010445393054539475, 7.00987669340838800450458919456, 7.999788693004321833563804908576, 8.606321959019546989558397165053, 8.888594894451070112717682271384, 10.26795190135486526030711682172