A note on uniqueness of $L$-functions in the extended Selberg class

The paper concerns uniqueness of $L$-functions in the extended Selberg class. It is shown that an $L$-function with positive degree is uniquely determined by the points of the set $E=\{\rho :L(\rho )=a_1$, or $a_2, \mathop{\rm Re} \rho \lt 0\}$ minus possibly a small set $G$, where $a_1$ and $a_2$ are two distinct constants. Some previous theorems given by Wu–Hu (2015) and Li–Du–Yi (2023) are generalized.