  Let ${\frak M}^\alpha$ be the spherical maximal operators of complex order
$\alpha$ on ${\mathbb R^n}$. In this article we show that when $n\geq 2$,
suppose \begin{eqnarray*} \|{\frak M}^{\alpha} f \|_{L^p({\mathbb R^n})} \leq
C\|f \|_{L^p({\mathbb R^n})} \end{eqnarray*} holds for some $\alpha$ and $p\geq
2$, then we must have ${\rm Re}\,\alpha \geq \max \{1/p-(n-1)/2,\ -(n-1)/p \}.$
When $n=2$, we prove that
  $\|{\frak M}^{\alpha} f \|_{L^p({\mathbb R^2})} \leq C\|f \|_{L^p({\mathbb
R^2})}$ if ${\rm Re}\ \ \alpha>\max\{1/p-1/2,\ -1/p\}$, and hence the range of
$\alpha$ is sharp in the sense the estimate fails for ${\rm Re}\ \alpha
<\max\{1/p-1/2, -1/ p\}.$
