Derive the Casimir Energy in Bosonic String Theory.
You start with the $\hat L_0$ operator and get rid of the non-vacuum $\displaystyle\frac{\alpha_0^2}{2}+\sum_{n=1}^\infty\alpha_{-n}\cdot\alpha_n$, then you use a Ramanujam sum to do $\zeta$-function renormalisation, from which you find out that the vacuum energy denoted by $\varepsilon_0$ is
$$\varepsilon_0=-\frac{d-2}{24}$$
However, the most interesting part comes when you go around deriving the critical dimension of Bosonic String Theory.
After which, the expression surprisingly simplifyies to a $-1$.
For a more detailed derivation of the above stuff, see these lecture notes/. (Section 4) (Equation 4.5-4.10)
