L(s) = 1 | + (0.275 + 0.158i)5-s + (−0.224 − 0.389i)7-s + (−4.22 + 2.43i)11-s + (−2.17 − 1.25i)13-s + 3·17-s + 4.24i·19-s + (−1.22 + 2.12i)23-s + (−2.44 − 4.24i)25-s + (5.72 − 3.30i)29-s + (3.44 − 5.97i)31-s − 0.142i·35-s − 9.43i·37-s + (−2.44 + 4.24i)41-s + (−0.674 + 0.389i)43-s + (−5.44 − 9.43i)47-s + ⋯ |
L(s) = 1 | + (0.123 + 0.0710i)5-s + (−0.0849 − 0.147i)7-s + (−1.27 + 0.735i)11-s + (−0.603 − 0.348i)13-s + 0.727·17-s + 0.973i·19-s + (−0.255 + 0.442i)23-s + (−0.489 − 0.848i)25-s + (1.06 − 0.613i)29-s + (0.619 − 1.07i)31-s − 0.0241i·35-s − 1.55i·37-s + (−0.382 + 0.662i)41-s + (−0.102 + 0.0593i)43-s + (−0.794 − 1.37i)47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2592 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.342 + 0.939i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2592 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.342 + 0.939i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.8395716643\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.8395716643\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
good | 5 | \( 1 + (-0.275 - 0.158i)T + (2.5 + 4.33i)T^{2} \) |
| 7 | \( 1 + (0.224 + 0.389i)T + (-3.5 + 6.06i)T^{2} \) |
| 11 | \( 1 + (4.22 - 2.43i)T + (5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (2.17 + 1.25i)T + (6.5 + 11.2i)T^{2} \) |
| 17 | \( 1 - 3T + 17T^{2} \) |
| 19 | \( 1 - 4.24iT - 19T^{2} \) |
| 23 | \( 1 + (1.22 - 2.12i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (-5.72 + 3.30i)T + (14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + (-3.44 + 5.97i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + 9.43iT - 37T^{2} \) |
| 41 | \( 1 + (2.44 - 4.24i)T + (-20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (0.674 - 0.389i)T + (21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (5.44 + 9.43i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + 9.75iT - 53T^{2} \) |
| 59 | \( 1 + (-4.89 - 2.82i)T + (29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (11.1 - 6.45i)T + (30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (9.67 + 5.58i)T + (33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 - 1.34T + 71T^{2} \) |
| 73 | \( 1 - 3.89T + 73T^{2} \) |
| 79 | \( 1 + (5.67 + 9.82i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (7.89 - 4.56i)T + (41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 - 11.6T + 89T^{2} \) |
| 97 | \( 1 + (6.44 + 11.1i)T + (-48.5 + 84.0i)T^{2} \) |
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
−8.443174777107702468887559178642, −7.83057353880826211567868872816, −7.34088015987438485535703172625, −6.24650725252728238556101961857, −5.53496671727150159379124022188, −4.75883993861480255166637409105, −3.82546434712740049661459726996, −2.74518039093718092143299400404, −1.93404960218474277075309273110, −0.27933647765130525790298770538,
1.24597992846829956495294813602, 2.69160818382893395338775316215, 3.15213201297118364796179545197, 4.59403933097800194107958183601, 5.13481625467921137705369376176, 5.99703032434563183604337451000, 6.83150347577493372481404957980, 7.66701994129269024590359203225, 8.339058494773218146802137981535, 9.062013797596896637150410980873