No, that is impossible. The $k$-th derivative of a L function has necessarily infinitely many zeros. So you can choose $s_j$ and $t_j$ inductively such that they give distinct zeros. Moreover, if one of $s_j$ is zero you can't say anything clever either, but I assume that you simply have forgotten that condition in your question.

No, that is impossible. The $k$-th derivative of a L function has necessarily infinitely many zeros. So you can choose $s_j$ and $t_j$ inductively such that the products give distinct zeros of $f^j$. Moreover, if one of $s_j$ is zero you can't say anything clever either, but I assume that you simply have forgotten that condition in your question.