Plate current use of 12Ax7 vacuum tube at low voltage
Looks like anode to cathode current will be in the 400uA range with zero grid voltage ("Idss"). 6AV6 has the same characteristics as the 12AX7 according to the databook. 



Edit: 

LTspice with the RCA12ax7 model yields a plate current of 417uA.

Model: 

.subckt RCA12ax7  1 2 3 *
*   *   *   *   *   *   *   *   *
eGIogVpc 20 0 value={log(v(1,3))}
rGlogVpc 20 0 1 
eG0 10 0 poly(1) &lt;2,3&gt;   -3.7694e+00   1.9947e+00   5.9432e-02    
eG1 11 0 poly(1) &lt;2,3&gt;   -3.2024e-02  -4.1443e-02  -4.8236e-03 
eG2 12 0 poly(1) &lt;2,3&gt;    1.9127e-02  -1.2189e-02  -1.5526e-03 
eG3 13 0 poly(1) &lt;2,3&gt;   -1.1354e-02   4.9339e-03   6.1016e-04  
rG0 10 0 1
rG1 11 0 1
rG2 12 0 1
rG3 13 0 1
gG   2 3 value={(exp(v(10)+v(20)*(v(11)+v(20)*(v(12)+v(20)*v(13)))))/170}
*
eP0 110 0 poly(1) &lt;2,3&gt;    -9.9158e+0  1.9145e+0 -2.8135e+0  1.8661e+0     
+                                                1.5643e+0  4.7240e-1  6.4276e-2  3.3101e-3 
eP1 111 0 poly(1) &lt;2,3&gt;     9.5428e-1  3.2558e-2 -8.3349e-1 -4.8578e-2  
+                                                2.6213e-1  1.0492e-1  1.8921e-2  1.3632e-3 
eP2 112 0 poly(1) &lt;2,3&gt;     9.5766e-2  2.5192e-2  2.2391e-1 -1.7040e-1 
+                                               -2.4952e-1 -1.0960e-1 -2.0981e-2 -1.4882e-3
eP3 113 0 poly(1) &lt;2,3&gt;    -6.6107e-2 -3.9657e-2  7.5560e-2  3.1025e-2  
+                                                2.4265e-2  1.7002e-2  4.2512e-3  3.4761e-4
eP4 114 0 poly(1) &lt;2,3&gt;     8.4148e-3  4.7989e-3 -1.3258e-2 -1.9288e-3  
+                                                5.2888e-4 -5.6853e-4 -2.4727e-4 -2.4359e-5
rP0 110 0 1
rP1 111 0 1
rP2 112 0 1
rP3 113 0 1
rP4 114 0 1
gP    1 3 value={(exp(v(110)+v(20)*(v(111)+v(20)*(v(112)+v(20)*(v(113)+v(20)*v(114))))))}
Cgc 2 3 1.8p
Cgp 2 1 1.7p
Cpc 1 3 1.9p
.ends


You can find some more info on this particular model here