We consider computing the longest palindrome in a text of length $n$ in the streaming model, where the characters arrive one-by-one, and we do not have random access to the input. While computing the answer exactly using sublinear memory is not possible in such a setting, one can still hope for a good approximation guarantee. We focus on the two most natural variants, where we aim for either additive or multiplicative approximation of the length of the longest palindrome. We first show that there is no point in considering Las Vegas algorithms in such a setting, as they cannot achieve sublinear space complexity. For Monte Carlo algorithms, we provide a lowerbound of $Ω(\frac{n}{E})$ bits for approximating the answer with additive error $E$, and $Ω(\frac{\log n}{\log(1+\varepsilon)})$ bits for approximating the answer with multiplicative error $(1+\varepsilon)$ for the binary alphabet. Then, we construct a generic Monte Carlo algorithm, which by choosing the parameters appropriately achieves space complexity matching up to a logarithmic factor for both variants. This substantially improves the previous results by Berenbrink et al. (STACS 2014) and essentially settles the space complexity.