Let $\{\lambda_n\}_{n\in\mathbb N}$  be a sequence of complex numbers .  Let's call a family of exponential functions $\{\exp (\lambda_n s)\}_{n\in\mathbb N}$ $F$-independent (where $F$ is either $\mathbb C$ or $\mathbb R$) iff whenever the series with complex coefficients

$$f(s)=\sum\limits_{n=1}^{\infty}a_n e^{\lambda_n s},\qquad s\in F,$$
converges to $f(s)\equiv 0$ uniformly on every compact subset of $F$, we have that 
$a_n=0$ for all $n\in\mathbb N$. 

**Question.** Assume that a sequence $\{\exp (\lambda_n s)\}_{n\in\mathbb N}$ is $\mathbb C$-independent. Is it $\mathbb R$-independent?