On the OP request, here is the plot of first 10000 partial sums.

[![enter image description here][1]][1]

Following Terry Tao's suggestion, here is the plot of ($n$th partial sum) $+2^{\frac32}/\sqrt{\pi n}$ for $n$ up to one million:

[![enter image description here][2]][2]

The thick line in the beginning actually consists of high frequency oscillations - in the range up to 2000 it looks like this:

[![enter image description here][3]][3]

(I hope there are no rounding artifacts, I calculated everything with 100 decimal digits precision)


  [1]: https://i.sstatic.net/beAmB.png
  [2]: https://i.sstatic.net/bWrPo.png
  [3]: https://i.sstatic.net/Dlq6M.png