Out of procrastination I computed $$\sum\_{k=1}^\infty k^{-k^2}\sim 1.06255080549625593786944593879.$$
The inverse symbolic calculator identified this number as "From generalized Zeta function". I do not recognize this sum as anything similar to a zeta function and obvious variations ($\sum\_{k=1}^\infty k^{-k}$ was recognized as what it is and $\sum\_{k=1}^\infty k^{-k^3}$ gave no result in the inverse symbolic calculator) seem not to be related to "generalized zeta functions". 


Does someone have a clue what "From generalized Zeta function" means?



What is a "generalized zeta function"?
A [General Dirichlet Series](http://en.wikipedia.org/wiki/General_Dirichlet_series) is a series of the form 
$$\sum\_{n=1}^{\infty} a\_ne^{-\lambda\_ns}$$
with complex $(a\_n)\_n$ and positive and strictly increasing $(\lambda\_n)\_n$.


A usual Dirichlet series is obtained for $\lambda\_n = \log n$ and then the Riemann $\zeta$-function for $a\_n=1$.


Your series is of this form at $s=1$, with $a\_n=1$ (so $\zeta$-like) and with $\lambda\_n = n^2 \log\_n$. Thus, to say it is a special value of a generalized $\zeta$-function makes some sense.


*However,* $\sum\_{k=1}^\infty k^{-k}$ and $\sum\_{k=1}^\infty k^{-k^3}$ would also be generalized $\zeta$-functions in this sense. Yet, arguably, it is plausible that the calculator recognizes the former as something more particular, and thus gives that information, and does not recognize the latter at all out of its limitations. 


