| L(s) = 1 | − 4·2-s + 6·3-s + 4·4-s + 10·5-s − 24·6-s + 24·7-s + 16·8-s + 9·9-s − 40·10-s − 50·11-s + 24·12-s − 200·13-s − 96·14-s + 60·15-s − 64·16-s − 34·17-s − 36·18-s + 46·19-s + 40·20-s + 144·21-s + 200·22-s − 126·23-s + 96·24-s + 25·25-s + 800·26-s − 54·27-s + 96·28-s + ⋯ |
| L(s) = 1 | − 1.41·2-s + 1.15·3-s + 1/2·4-s + 0.894·5-s − 1.63·6-s + 1.29·7-s + 0.707·8-s + 1/3·9-s − 1.26·10-s − 1.37·11-s + 0.577·12-s − 4.26·13-s − 1.83·14-s + 1.03·15-s − 16-s − 0.485·17-s − 0.471·18-s + 0.555·19-s + 0.447·20-s + 1.49·21-s + 1.93·22-s − 1.14·23-s + 0.816·24-s + 1/5·25-s + 6.03·26-s − 0.384·27-s + 0.647·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{4} \cdot 5^{4} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{4} \cdot 5^{4} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.364894247\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.364894247\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | $C_2$ | \( ( 1 + p T + p^{2} T^{2} )^{2} \) |
| 3 | $C_2$ | \( ( 1 - p T + p^{2} T^{2} )^{2} \) |
| 5 | $C_2$ | \( ( 1 - p T + p^{2} T^{2} )^{2} \) |
| 7 | $C_2^2$ | \( 1 - 24 T + 535 T^{2} - 24 p^{3} T^{3} + p^{6} T^{4} \) |
| good | 11 | $D_4\times C_2$ | \( 1 + 50 T - 492 T^{2} + 1500 p T^{3} + 31843 p^{2} T^{4} + 1500 p^{4} T^{5} - 492 p^{6} T^{6} + 50 p^{9} T^{7} + p^{12} T^{8} \) |
| 13 | $D_{4}$ | \( ( 1 + 100 T + 6599 T^{2} + 100 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 17 | $D_4\times C_2$ | \( 1 + 2 p T - 1584 T^{2} - 14172 p T^{3} - 22309397 T^{4} - 14172 p^{4} T^{5} - 1584 p^{6} T^{6} + 2 p^{10} T^{7} + p^{12} T^{8} \) |
| 19 | $D_4\times C_2$ | \( 1 - 46 T - 7411 T^{2} + 192786 T^{3} + 29204204 T^{4} + 192786 p^{3} T^{5} - 7411 p^{6} T^{6} - 46 p^{9} T^{7} + p^{12} T^{8} \) |
| 23 | $D_4\times C_2$ | \( 1 + 126 T + 11468 T^{2} - 2510676 T^{3} - 321768717 T^{4} - 2510676 p^{3} T^{5} + 11468 p^{6} T^{6} + 126 p^{9} T^{7} + p^{12} T^{8} \) |
| 29 | $D_{4}$ | \( ( 1 + 150 T + 51748 T^{2} + 150 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 31 | $D_4\times C_2$ | \( 1 - 74 T - 12995 T^{2} + 3042214 T^{3} - 709461356 T^{4} + 3042214 p^{3} T^{5} - 12995 p^{6} T^{6} - 74 p^{9} T^{7} + p^{12} T^{8} \) |
| 37 | $D_4\times C_2$ | \( 1 - 176 T - 54179 T^{2} + 2842576 T^{3} + 3116620288 T^{4} + 2842576 p^{3} T^{5} - 54179 p^{6} T^{6} - 176 p^{9} T^{7} + p^{12} T^{8} \) |
| 41 | $D_{4}$ | \( ( 1 - 10 T + 52612 T^{2} - 10 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 43 | $D_{4}$ | \( ( 1 - 132 T + 148915 T^{2} - 132 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 47 | $D_4\times C_2$ | \( 1 + 316 T - 128034 T^{2} + 6397104 T^{3} + 30787287283 T^{4} + 6397104 p^{3} T^{5} - 128034 p^{6} T^{6} + 316 p^{9} T^{7} + p^{12} T^{8} \) |
| 53 | $D_4\times C_2$ | \( 1 + 236 T - 226482 T^{2} - 3675936 T^{3} + 49168209163 T^{4} - 3675936 p^{3} T^{5} - 226482 p^{6} T^{6} + 236 p^{9} T^{7} + p^{12} T^{8} \) |
| 59 | $D_4\times C_2$ | \( 1 - 58 T + 300060 T^{2} + 41032332 T^{3} + 45157087819 T^{4} + 41032332 p^{3} T^{5} + 300060 p^{6} T^{6} - 58 p^{9} T^{7} + p^{12} T^{8} \) |
| 61 | $D_4\times C_2$ | \( 1 - 792 T + 21206 T^{2} - 120460032 T^{3} + 173408915019 T^{4} - 120460032 p^{3} T^{5} + 21206 p^{6} T^{6} - 792 p^{9} T^{7} + p^{12} T^{8} \) |
| 67 | $D_4\times C_2$ | \( 1 - 868 T + 13397 T^{2} - 120218868 T^{3} + 230579516048 T^{4} - 120218868 p^{3} T^{5} + 13397 p^{6} T^{6} - 868 p^{9} T^{7} + p^{12} T^{8} \) |
| 71 | $D_{4}$ | \( ( 1 + 386 T + 568696 T^{2} + 386 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 73 | $D_4\times C_2$ | \( 1 - 1220 T + 404641 T^{2} - 372984500 T^{3} + 444186440992 T^{4} - 372984500 p^{3} T^{5} + 404641 p^{6} T^{6} - 1220 p^{9} T^{7} + p^{12} T^{8} \) |
| 79 | $D_4\times C_2$ | \( 1 + 54 T - 412771 T^{2} - 30801114 T^{3} - 71729475276 T^{4} - 30801114 p^{3} T^{5} - 412771 p^{6} T^{6} + 54 p^{9} T^{7} + p^{12} T^{8} \) |
| 83 | $D_{4}$ | \( ( 1 + 182 T + 54160 T^{2} + 182 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 89 | $D_4\times C_2$ | \( 1 - 418 T - 1122840 T^{2} + 46972332 T^{3} + 1063516039579 T^{4} + 46972332 p^{3} T^{5} - 1122840 p^{6} T^{6} - 418 p^{9} T^{7} + p^{12} T^{8} \) |
| 97 | $D_{4}$ | \( ( 1 + 392 T + 1821282 T^{2} + 392 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.447297496274576150933458403723, −8.444757614506127608378020084009, −7.957100605783426680182495640876, −7.82475159170796428386663701099, −7.60156812747363580014502578183, −7.40550340846519955359493379448, −7.36642397125505176918646889983, −6.87680672368604905631711647785, −6.61916243599681186708255808427, −5.95771697167499847451237349937, −5.76720952491935567303092537101, −5.33966141220376589050668163248, −5.16813493198484130569745212607, −4.79634137783997384358518690071, −4.68426459321033210576275555923, −4.40544537623334601670972290640, −3.79477233171524835756394076682, −3.36025165098572157238674815081, −2.68897573478312415297766159538, −2.61293865470010310172286477694, −2.10339035321407649469355062346, −1.99826675736112997550102117907, −1.78043317063670195835673674745, −0.56793223213623270083074822972, −0.40822855465790036132351985741,
0.40822855465790036132351985741, 0.56793223213623270083074822972, 1.78043317063670195835673674745, 1.99826675736112997550102117907, 2.10339035321407649469355062346, 2.61293865470010310172286477694, 2.68897573478312415297766159538, 3.36025165098572157238674815081, 3.79477233171524835756394076682, 4.40544537623334601670972290640, 4.68426459321033210576275555923, 4.79634137783997384358518690071, 5.16813493198484130569745212607, 5.33966141220376589050668163248, 5.76720952491935567303092537101, 5.95771697167499847451237349937, 6.61916243599681186708255808427, 6.87680672368604905631711647785, 7.36642397125505176918646889983, 7.40550340846519955359493379448, 7.60156812747363580014502578183, 7.82475159170796428386663701099, 7.957100605783426680182495640876, 8.444757614506127608378020084009, 8.447297496274576150933458403723
