| L(s) = 1 | + 12·9-s + 144·29-s + 1.44e3·41-s + 1.48e3·49-s + 3.55e3·61-s + 2.36e3·81-s − 1.80e3·89-s + 4.06e3·101-s − 5.40e3·109-s + 1.31e3·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 5.89e3·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + ⋯ |
| L(s) = 1 | + 4/9·9-s + 0.922·29-s + 5.51·41-s + 4.31·49-s + 7.45·61-s + 3.24·81-s − 2.14·89-s + 4.00·101-s − 4.75·109-s + 0.988·121-s + 0.000698·127-s + 0.000666·131-s + 0.000623·137-s + 0.000610·139-s + 0.000549·149-s + 0.000538·151-s + 0.000508·157-s + 0.000480·163-s + 0.000463·167-s + 2.68·169-s + 0.000439·173-s + 0.000417·179-s + 0.000410·181-s + 0.000378·191-s + 0.000372·193-s + 0.000361·197-s + 0.000356·199-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{40} \cdot 5^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{40} \cdot 5^{16}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.9376314701\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9376314701\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| good | 3 | \( ( 1 - 2 p T^{2} - 1129 T^{4} - 2 p^{7} T^{6} + p^{12} T^{8} )^{2} \) |
| 7 | \( ( 1 - 740 T^{2} + 330662 T^{4} - 740 p^{6} T^{6} + p^{12} T^{8} )^{2} \) |
| 11 | \( ( 1 - 658 T^{2} + 1468127 T^{4} - 658 p^{6} T^{6} + p^{12} T^{8} )^{2} \) |
| 13 | \( ( 1 - 2948 T^{2} + 8461878 T^{4} - 2948 p^{6} T^{6} + p^{12} T^{8} )^{2} \) |
| 17 | \( ( 1 - 14010 T^{2} + 95008763 T^{4} - 14010 p^{6} T^{6} + p^{12} T^{8} )^{2} \) |
| 19 | \( ( 1 + 23046 T^{2} + 226806391 T^{4} + 23046 p^{6} T^{6} + p^{12} T^{8} )^{2} \) |
| 23 | \( ( 1 - 2276 T^{2} + 282372326 T^{4} - 2276 p^{6} T^{6} + p^{12} T^{8} )^{2} \) |
| 29 | \( ( 1 - 36 T + 25738 T^{2} - 36 p^{3} T^{3} + p^{6} T^{4} )^{4} \) |
| 31 | \( ( 1 + 38852 T^{2} + 1997821382 T^{4} + 38852 p^{6} T^{6} + p^{12} T^{8} )^{2} \) |
| 37 | \( ( 1 - 48236 T^{2} + 682954998 T^{4} - 48236 p^{6} T^{6} + p^{12} T^{8} )^{2} \) |
| 41 | \( ( 1 - 362 T + 129067 T^{2} - 362 p^{3} T^{3} + p^{6} T^{4} )^{4} \) |
| 43 | \( ( 1 - 236220 T^{2} + 26099208982 T^{4} - 236220 p^{6} T^{6} + p^{12} T^{8} )^{2} \) |
| 47 | \( ( 1 - 262380 T^{2} + 38440240102 T^{4} - 262380 p^{6} T^{6} + p^{12} T^{8} )^{2} \) |
| 53 | \( ( 1 - 248470 T^{2} + p^{6} T^{4} )^{4} \) |
| 59 | \( ( 1 + 242748 T^{2} + 98833488982 T^{4} + 242748 p^{6} T^{6} + p^{12} T^{8} )^{2} \) |
| 61 | \( ( 1 - 888 T + 648502 T^{2} - 888 p^{3} T^{3} + p^{6} T^{4} )^{4} \) |
| 67 | \( ( 1 - 645470 T^{2} + 282431706287 T^{4} - 645470 p^{6} T^{6} + p^{12} T^{8} )^{2} \) |
| 71 | \( ( 1 + 1003212 T^{2} + 461928439942 T^{4} + 1003212 p^{6} T^{6} + p^{12} T^{8} )^{2} \) |
| 73 | \( ( 1 + 84870 T^{2} + 93725642203 T^{4} + 84870 p^{6} T^{6} + p^{12} T^{8} )^{2} \) |
| 79 | \( ( 1 + 782308 T^{2} + 322542408902 T^{4} + 782308 p^{6} T^{6} + p^{12} T^{8} )^{2} \) |
| 83 | \( ( 1 - 347726 T^{2} + 585889946111 T^{4} - 347726 p^{6} T^{6} + p^{12} T^{8} )^{2} \) |
| 89 | \( ( 1 + 450 T + 1395663 T^{2} + 450 p^{3} T^{3} + p^{6} T^{4} )^{4} \) |
| 97 | \( ( 1 - 3099740 T^{2} + 4051555388358 T^{4} - 3099740 p^{6} T^{6} + p^{12} T^{8} )^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{16} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−3.98580960367699463716697391757, −3.77394570633646915210492292276, −3.71963960621793145932268036255, −3.64292462135127009382898350380, −3.59234836999917873547731093525, −3.56099518605140875837554258719, −3.01754085758991043333289689593, −2.87020620664953497978229698944, −2.81609764578195789641969138820, −2.70839727807963768486179180653, −2.54713781940208736037664674209, −2.33361115477390013944692328700, −2.22510618573877227689657545658, −2.21669384813536641245880321668, −2.20846302737937141140596658127, −1.83840139639787994774222315789, −1.65217422339950666728788142448, −1.26665333210474088595075721804, −1.04268803124050707868490558317, −1.02078561666735844725293145893, −0.881611301171309135062116763455, −0.76430745300423087283678873326, −0.74814420100568060424759495725, −0.43075505194852935755442501143, −0.04025275624678756945596929908,
0.04025275624678756945596929908, 0.43075505194852935755442501143, 0.74814420100568060424759495725, 0.76430745300423087283678873326, 0.881611301171309135062116763455, 1.02078561666735844725293145893, 1.04268803124050707868490558317, 1.26665333210474088595075721804, 1.65217422339950666728788142448, 1.83840139639787994774222315789, 2.20846302737937141140596658127, 2.21669384813536641245880321668, 2.22510618573877227689657545658, 2.33361115477390013944692328700, 2.54713781940208736037664674209, 2.70839727807963768486179180653, 2.81609764578195789641969138820, 2.87020620664953497978229698944, 3.01754085758991043333289689593, 3.56099518605140875837554258719, 3.59234836999917873547731093525, 3.64292462135127009382898350380, 3.71963960621793145932268036255, 3.77394570633646915210492292276, 3.98580960367699463716697391757
Plot not available for L-functions of degree greater than 10.
