MacLaurin expansion of even functions

[Jacques Gélinas <jacquesg00@hotmail.com>]

$ \p
   realprecision = 57 significant digits (5 digits displayed)
$ \ps 4
   seriesprecision = 4 significant terms

Consider the differences between the first two and last two expansions

$ Vec(besselj(-1/2,t))           \\ cos
[1, 0, -1/2, 0, 1/24]
$ Vec(besselj(1/2,t))             \\ sinc
[1, 0, -1/6, 0, 1/120]

$ xis(s) = gamma(1+s/2)/Pi^(s/2)*(s-1)*zeta(s);
$ Vec(xis(1/2+t))
[0.49712, 0.E-57, 0.011486, -1.2745 E-57]
$ Xi(t) = lfunlambda(1,1/2+t) * binomial(1/2+t,2);
$ Vec(Xi(t))
[0.49712, 0.E-77, 0.011486, 0.E-76]

Of course, I would want exact zeros for xis, Xi as for the Bessel functions,
and, if possible, the same number of coefficients for a given series precision.

How can this be done simply for any even function ? 
Sereven(f,var) = ???

Jacques Gélinas