analytic number theory - Uniform variant of Stirling's approximation

Stirling's formula is usually stated in the form $log Gamma(s) = (s-frac12) log{s} - s + logsqrt{2pi} + E(s)$, where
$E(s) = c_1/s + c_2/s^2 + dots + O(s^{-K})$ for certain absolute constants $c_i$. I am interested in having a uniform approximation for $E(s)$ that is valid for all $s = sigma + it$ with $sigma>0$ fixed and $|t| leq X$ for $X geq 1$. Does there exist a known "nice" approximation for $E(s)$ of the form $E(s) = F(s) + O(X^{-K})$, where
$F(s)$, which depends on $K$ and $X$ of course, has an explicit shape? Bonus points for explicit bounds
on $F(s)$ and its derivatives uniformly valid in $|t| leq X$.

EDIT ADDED July 28 2010: I am doubtful if there is a positive answer to my question. As a simple example, consider the rate of convergence of the Taylor series of the cosine function. Of course, $cos(x) = 1- frac{x^2}{2!} + frac{x^4}{4!} - dots pm frac{x^{2k}}{(2k)!} + R_{2k}(x)$ where $R_{2k}(x) = cos^{(2k+1)}(xi) frac{x^{2k+1}}{(2k+1)!}$ for some $|xi| leq |x|$. In order to get an error term that is $O(X^{-K})$ uniformly for $|x| leq X$ we need to take $k$ roughly on the order of $X$ (since that is when the factorial in the denominator wins over the size of $X^{2k+1}$); at this point the error term gets very small very fast . This is a lot of terms!