Let $$L(C,s)=\sum_{n=1}^\infty \frac{a_n}{n^s}$$ be the Dirichlet series of the Hasse--Weil L-function of an elliptic curve $C$ over $ℚ$. As $s$ takes on real negative values, there are trivial zeros at the integers. The sign of $L$ changes as we pass through a zero. We can also check this property:

(P) $|L(-n+1/2)|$ gets large as $n$ gets large.

My question is: Does the property (P) holds true for the derivatives of $L$, namely, $L^{(k)},k=1,2,...$