(* You can copy and paste this into your own Mathematica code.  I generated the following array of zeroes of BesselJPrime functions via the "BesselZeros.m" package, downloadable from http://library.wolfram.com/infocenter/MathSource/6777/. Works for 0 ≤ α ≤ 10 and 1 ≤ m ≤ 10. If you want more zeroes than what I've given here, you should download the package yourself and use the BesselJPrimeZeros function it provides. *)
besselprimezeroesarray={{3.831705970207512,7.015586669815622,10.173468135062722,13.323691936314223,16.470630050877634,19.615858510468243,22.760084380592772,25.903672087618382,29.046828534916855,32.189679910974405},{1.841183781340659,5.331442773525033,8.536316366346233,11.706004902592063,14.863588633909032,18.015527862681804,21.164369859188792,24.311326857210776,27.457050571059245,30.601922972669094},{3.054236928227141,6.706133194158456,9.969467823087555,13.170370856016122,16.347522318321783,19.512912782488204,22.671581772477424,25.82603714178526,28.977672772993678,32.127327020443474},{4.201188941210529,8.01523659837596,11.345924310742806,14.585848286167028,17.788747866066473,20.9724769365377,24.144897432909264,27.310057930204348,30.470268806290424,33.62694918279668},{5.317553126083995,9.282396285241525,12.68190844263889,15.96410703773155,19.196028800048904,22.401032267689004,25.58975968138673,28.767836217666503,31.93853934097278,35.10391667734676},{6.415616375700238,10.519860873772286,13.9871886301403,17.312842487884627,20.57551452138689,23.803581476593862,27.01030789777772,30.20284907898166,33.38544390101012,36.56077768688036},{7.501266144684147,11.73493595304269,15.268181461097873,18.6374430096662,21.931715017802237,25.183925599499624,28.409776362510083,31.617875716105033,34.81339298429743,37.9996408977153},{8.577836489714072,12.93238623708888,16.529365884366943,19.941853366527344,23.26805292645757,26.545032061823576,29.790748583196613,33.015178641375144,36.22438054878716,39.42227457893926},{9.647421651997213,14.115518907894602,17.774012366915255,21.22906262285312,24.587197486317677,27.889269427955092,31.155326556188324,34.396628554272176,37.620078044197086,40.83017868182204},{10.711433970699952,15.286737667333028,19.004593537946054,22.501398726777282,25.891277276839137,29.21856349993608,32.50524735237553,35.7637929288088,39.00190281151422,42.22463843075328},{11.770876674955558,16.4478527484865,20.2230314126817,23.760715860327448,27.182021527190532,30.534504754007074,33.84196577513571,37.1180004236656,40.37106890533389,43.60676490137951}};
besselprimezeroes[α_,m_]:=besselprimezeroesarray[[α+1,m]]   (* the alpha+1 is needed because the Bessel functions start at 0 whereas arrays in Mathematica start at 1 *)

besselprimezeroes[2,4] (*Example  besselprimezeroes[2,4] gives the 4 th zero of the J2'[x] function *)