By analyzing the graph of the function $\zeta \left(\frac{1}{2}+it\right)$ I've noticed a circular shape for $-3.3<t<3.3$ as shown below.
Based on that, an approximation for such function on the mentioned interval is $$\zeta \left(\frac{1}{2}+it\right) \approx -\left(\zeta \left(\frac{1}{2}\right) +\frac{1}{2}\right)e^{it}-\frac{1}{2}$$
Is there a way to derive such approximation algebraically from the analytic continued equation of the Riemann zeta function?