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 $m$. The sign of $L$ changes as we pass through a zero. We can also check this property:

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

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

More generally, If $L(C,s)$ assumes arbitrarily large and arbitrarily small values, i.e., for all $K>0$, there are $a,b$ with $L(C,a)<−K$ and $L(C,b)>K$. Does this property holds true for its derivatives?