@Markiyan Hirnyk By the way, I vastly underestimated the number of evaluations I need to make in (say) an hour or so.  One thing I do is visualize Julia sets, so I need to compute the forward orbit of points under iteration of an L-function.  So for each of the 2.5*10^5 values of s I contemplate examining, I might need to do up to  10 or 20 evaluations of its forward orbit as well.

All this is practical for zeta, in my Mathematica experience.  Not so for the slow Ramanujan L-series command.

Barry

@Axel Vogt 

 Point taken; I'll stick to the list.

Barry

@Markiyan Hirnyk 

Marlyan, not sure this explains it.  One can quickly make large tables of the Ramanujan tau function, for example (Fourier coefficients of the Ramanujan tau-Dirichlet series) using Lehmer's method.  I think the reason computing this L-function must lie deeper.

Barry

@Axel Vogt 

Thanks Axel, I'll look into this too.  Will you send me your email address?  I might want to ask you questions about invoking PARI from Maple.

And what does "SW" mean?

Barry

@Markiyan Hirnyk 

Markiyan, hi.  Maybe I'll have to live with this. but it isn't  as fast as I'd like.  I want to do be able to do this many times in a given session, to produce graphical representations of these functions, not for display, but as experiments.  I'd like to be able to follow my nose and repeat the visualizations with variations as the data suggests in real time. Mathematica performs quickly when asked to do this with the Riemann zeta function--maybe that experience has spoiled me.  I wonder, is there a theoretical reason it just takes longer to compute other L-functions?  Or not.

Also my system seems to be about 50% slower than yours:

In[7]:= Timing[RamanujanTauL[6 + 9.22 I]]

Out[7]= {0.067076, 0.00040309 - 0.00239013 I}

Thanks, though.

Barry

@Markiyan Hirnyk 

Thanks. But as I said I want to compute 2.5*10^5 values in a reasonable time, and Mathematica's routine executes too slowly to make this practical.  I'm making graphics and I'd like to have this level of performance to achieve good resolution (500 by 500 pixels.)

Even if Maple has no built-in function to compute this series, maybe someone has written a Maple routine to do it (faster than Mathematica's)? 

Barry Brent