I'm not entirely sure what you mean by your question. Here are two remarks:

If you *assume* BSD, then to "construct" $L_E$ you just need to give the curve $E$. There are (many) elliptic curves /$\mathbb{Q}$ whose ranks have been computed, and are (say) equal to 3 or 4.

If one wants an example without assuming BSD, then you are in trouble - for given $E$, you can compute $L^{(n)}(s)$ to any desired degree of accuracy, but proving that it vanishes computationally is impossible.

However, two things help you. If the sign in the functional equation is -1, then you have that the order of vanishing is also odd. The Gross-Zagier formula can be used to check the vanishing of the first derivative. For example, this is used in the following paper to exhibit an elliptic curve $E$ whose $L_E$ provably vanishes to order 3.

On the Conjecture of Birch and Swinnerton-Dyer for an Elliptic Curve of Rank 3
Author(s): Joe P. Buhler, Benedict H. Gross, Don B. Zagier
Source: Mathematics of Computation, Vol. 44, No. 170 (Apr., 1985), pp. 473-481