This post builds on an MSE question about a conjectured series expression for the Riemann $\xi$-function:
$$\xi(s) = \xi(1-s) = \sum_{n=1}^\infty (-1)^{n+1}\,\big(\xi\left(s+in\right)+\xi\left(1-s+in\right)\big) \qquad s \in \mathbb{C} \tag{1}$$
Numerical evidence suggests this series could be generalised to include all Completed Dirichlet L-series:
$$\Lambda(s,\chi) = \epsilon\, \Lambda(1-s,\overline{\chi}) = \sum_{n=1}^\infty (-1)^{n+1}\,\bigg(\Lambda\left(s+in,\chi\right)+\epsilon\,\overline{\Lambda\left(\overline{1-s+in},\chi\right)}\bigg) \qquad s \in \mathbb{C} \tag{2}$$
where $\epsilon$ is the sign or the root number. The pari/gp code below could be used to verify the observation:
\\Test for Dirichlet L-series (primitive characters only)
default(realprecision,30)
p = 22; q = 23; \\p = Conrey index, q = modulus, (p,q) need to be coprime.
L =lfuncreate(Mod(p,q));
eps=lfunrootres(L)[3];
Lamb(s) = if(q==1, s/2*(s-1)*lfunlambda(L,s), lfunlambda(L,s));
Lamb1ms(s)=eps*conj(Lamb(conj(1-s)));
LambSum(s,v)=sum(n=1,v,(-1)^(n+1)*(Lamb(s+I*n)+eps*conj(Lamb(conj(1-s+I*n)))));
val = 12+I; \\pick the desired value for s
print("Check functional equation:")
print(Lamb(val));
print(Lamb1ms(val));
print("The series expression:")
print(LambSum(val,100));
I have tried a couple of other entire functions with a similar functional equation like ${}_2F_0 \left([s, 1-s],[], -\frac12\right)$ or $\frac{1}{\Gamma(s)}+\frac{1}{\Gamma(1-s)}$, however none of these comply.
It might be a specific attribute of the completed Dirichlet L-series (or a series with a similar speed of decay). Therefore also tested the Dirichlet $\eta$-function and since this does work, the phenomenon does not seem to be restricted to the Selberg-class.
The initial idea for a proof of the conjecture was to apply the alternating Abel Plana formula. Although this initially looked promising, from a sequence of comments to the MSE question, it has become clear this approach will likely fail.
Question:
The numerical evidence for the conjecture is quite strong. Grateful for any suggestions about possible paths that might lead towards a proof (or disproof).
Best Answer
Your conjecture is true, and follows trivially from the Poisson summation formula. For instance, use Theorem 9.4.2 in my book joint with K. Belabas "Numerical Algorithms for number theory", GSM 254, AMS (sorry for the self-promotion) applied to $h=2\pi$, subtract the formula for $s$ and for $s+i$ gives your alternating sum and everything cancels. There is certainly a faster method using an alternating version of Poisson (note that $h=2\pi$ is essential since one uses $\exp(ihm)=1$).