Why do the first spikes in these plots point in opposite directions? With the following Mathematica program:
Clear[x, n, nn]
nn = Prime[13] + 1;
A = -Sum[Re[x^N[ZetaZero[n]]], {n, 1, 50}];
Plot[A, {x, 0, nn}, PlotRange -> {-80, 120}]

I plotted the function:
$$f(x)=-\sum _{n=1}^{50} \Re\left(x^{\rho _n}\right)$$
Where $\rho _n$ is the $n$-th Riemann zeta zero.

With this second program:
Clear[n, k, x, nn]
nn = Prime[13] + 1;
Monitor[A = 
   Sum[Sum[Re[Log[Prime[n]]*x^(2 I \[Pi]*k/Log[Prime[n]])], {n, 1, 
      13}], {k, 1, 2*nn}];, k]
Plot[A, {x, 0, nn}, PlotRange -> {-200, 400}]

I plotted the function:
$$g(x) = \sum _{k=1}^{2 \text{nn}} \left(\sum _{n=1}^{13} \Re\left(\log (p_n) x^{\frac{2 i \pi  k}{\log (p_n)}}\right)\right)$$
where $p_n$ is the $n$-th prime number.

In the first picture there is a spike at $x=1$ by the pointer, pointing down.
In the second picture there is a spike at $x=1$ by the pointer, pointing up.
Is there a simple explanation to this? And can these first spikes in the plots be made to point in the same direction without changing the rest of the wave?

The spikes occur at the value $x=1$ and can be easily checked using exact arithmetic.  For the first one, in fact, since $1^z=1$ for any $z$ we easily see that the value should be $-50$, as it appears in your graph.  For the second, we get
Clear[n, k, x, nn]
nn = Prime[13] + 1;
A = Sum[Sum[
  Re[Log[Prime[n]]*x^(2 I \[Pi]*k/Log[Prime[n]])], {n, 1, 13}], {k, 1, 2*nn}];
A /. x -> 1

(* Out: 
  84 Log[2] + 84 Log[3] + 84 Log[5] + 84 Log[7] + 84 Log[11] + 
    84 Log[13] + 84 Log[17] + 84 Log[19] + 84 Log[23] + 84 Log[29] + 
    84 Log[31] + 84 Log[37] + 84 Log[41]
*)

N[%]

(* Out: 2801.31 *)

All looks OK to me.
