N coins have probability $p_n = e^{-t_n/s}$ of heads, $t_n$ being specific for each coin. Coins 1 to m came up heads and m+1 to N came up tails. Now I'm trying to estimate $s$ using the Maximum Likelihood Method.

$L(s) = p_1 p_2 \dots p_m (1-p_{m+1})\dots(1-p_N)$

But this function is difficult to maximize. Do I have to resort to numerical methods?