More or less by definition, $\tau(\lambda)$ is the ratio of the periods of $E(\lambda):y^2=x(x-1)(x-\lambda)$. The periods are the integrals of $dx/y$ over a pair of generators for $H_1(E_n, \mathbf{Z})$. Ambiguity in the choice of generators for the homology of $E$ implies that $\tau(\lambda)$ is only well-defined up to the action of an element of $\text{SL}_2(\mathbf{Z})$.
The periods are given, for example, by
$\left\{2\int_0^1 \frac{\mathrm d x}{y}, 2\int_1^\lambda \frac{\mathrm dx}{y}\right\}.$
These can be expressed in terms of the hypergeometric function using Euler's integral representation of the hypergeometric function.
Finding special values of $\tau$, I suppose, is then equivalent to finding special values of the hypergeometric function. You can use a CAS to get an arbitrarily good approximation, but I'm in no position to give an exact value of your $\tau(E_n)$ as a function of $n$.
Where did you get this problem from?