Let \(S^*_ \psi(A)\) be the class of entire Dirichlet series \[ F(s)= \sum^ \infty_{n=1} a_ n \exp\lambda n^ s,\quad s= \sigma+ it,\quad A= \{a_ n\} \] is a decreasing sequence with \(\sum^ \infty_{n=1} a_ n< +\infty\) and \(\lambda_ n\nearrow\infty\) as \(n\to\infty\) and \(\lambda_ n\leq {\log^ +(1/a_ n)\over \psi\log(1/a_ n)}\), where \(\psi\) is a positive continuous function on \([0,\infty]\) with \(\psi(x)\uparrow +\infty\) and \({x\over \psi(x)}\to \infty\) as \(x\to\infty\). Then \(\psi\) is said to be in \(L^*\). The main result of the paper gives the necessary and sufficient condition for \(\psi(\log M(\sigma,F))\sim\varphi(\log\mu(\sigma, F))\), where \[ M(\sigma,F)= \{\sup| F(\sigma+ it)| t\in \mathbb{R}\}, \] \(\mu(\sigma,F)= \max\{a_ n\exp\sigma\lambda_ n\}\), \(n\in \mathbb{Z}_ n\), \(\varphi\) is a positive increasing continuous function for which \(\log\varphi(x)\) is concave and \(\varphi(\log x)\) is a slowly increasing function. Let \(\psi\in L^*\) and \(\limsup_{n\to\infty} {\log n\over A_ n}= h<1\). Let \(\varphi\in L\) and \(\varphi(2x)\sim\varphi(x)\) as \(n\to\infty\) and \(\log\varphi(x)\) is concave. Then \(\varphi(\log M(\sigma,F))\sim\varphi(\log\mu(\sigma, F))\), \(\sigma\to\infty\) holds for \(F\in S^*_ \psi(A)\) if and only if \[ \lim_{n\to\infty}= {\varphi(\log n+\psi(A_ n)\gamma(\gamma(A_ n))\over \varphi(\psi(A_ n))\gamma(\psi(A_ n)))}= 1 \] for any function \(\gamma\in L\) and \(\gamma(2x)\sim\gamma(x)\), \(x\to\infty\).