We construct two Banach algebras, one which contains analytic semigroups ${\left(a^z\right)}_{Rez>0}$ such that $|a^{1+iy}|\to \infty$ arbitrarily slowly as $\left|y\right|\to \infty$, the other which contains ones such that $|a^{1+iy}|\to \infty$ arbitrarily fast