\[\zeta'\left(-1,a\right)-\frac{1}{12}+\frac{1}{4}a^{2}-\left(\frac{1}{12}-\frac%
{1}{2}a+\frac{1}{2}a^{2}\right)\ln a\sim-\sum_{k=1}^{\infty}\frac{B_{2k+2}}{(2%
k+2)(2k+1)2k}a^{-2k},\]