| L(s) = 1 | − 4·2-s + 4·4-s − 12·5-s + 16·8-s + 48·10-s − 4·11-s − 96·13-s − 64·16-s − 84·17-s − 72·19-s − 48·20-s + 16·22-s − 308·23-s + 188·25-s + 384·26-s + 160·29-s + 384·31-s + 64·32-s + 336·34-s − 536·37-s + 288·38-s − 192·40-s + 1.51e3·41-s + 800·43-s − 16·44-s + 1.23e3·46-s − 312·47-s + ⋯ |
| L(s) = 1 | − 1.41·2-s + 1/2·4-s − 1.07·5-s + 0.707·8-s + 1.51·10-s − 0.109·11-s − 2.04·13-s − 16-s − 1.19·17-s − 0.869·19-s − 0.536·20-s + 0.155·22-s − 2.79·23-s + 1.50·25-s + 2.89·26-s + 1.02·29-s + 2.22·31-s + 0.353·32-s + 1.69·34-s − 2.38·37-s + 1.22·38-s − 0.758·40-s + 5.75·41-s + 2.83·43-s − 0.0548·44-s + 3.94·46-s − 0.968·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{8} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{8} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.7198067855\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7198067855\) |
| \(L(\frac{5}{2})\) |
|
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 | | \( 1 \) |
| 7 | | \( 1 \) |
| good | 5 | $D_4\times C_2$ | \( 1 + 12 T - 44 T^{2} - 744 T^{3} + 1719 T^{4} - 744 p^{3} T^{5} - 44 p^{6} T^{6} + 12 p^{9} T^{7} + p^{12} T^{8} \) |
| 11 | $D_4\times C_2$ | \( 1 + 4 T + 878 T^{2} - 14096 T^{3} - 1049813 T^{4} - 14096 p^{3} T^{5} + 878 p^{6} T^{6} + 4 p^{9} T^{7} + p^{12} T^{8} \) |
| 13 | $D_{4}$ | \( ( 1 + 48 T + 4088 T^{2} + 48 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 17 | $D_4\times C_2$ | \( 1 + 84 T - 4436 T^{2} + 8232 p T^{3} + 253503 p^{2} T^{4} + 8232 p^{4} T^{5} - 4436 p^{6} T^{6} + 84 p^{9} T^{7} + p^{12} T^{8} \) |
| 19 | $D_4\times C_2$ | \( 1 + 72 T - 9438 T^{2} + 65088 T^{3} + 131199947 T^{4} + 65088 p^{3} T^{5} - 9438 p^{6} T^{6} + 72 p^{9} T^{7} + p^{12} T^{8} \) |
| 23 | $D_4\times C_2$ | \( 1 + 308 T + 50342 T^{2} + 6217904 T^{3} + 679962307 T^{4} + 6217904 p^{3} T^{5} + 50342 p^{6} T^{6} + 308 p^{9} T^{7} + p^{12} T^{8} \) |
| 29 | $D_{4}$ | \( ( 1 - 80 T + 18626 T^{2} - 80 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 31 | $D_4\times C_2$ | \( 1 - 384 T + 54538 T^{2} - 12801024 T^{3} + 3353389347 T^{4} - 12801024 p^{3} T^{5} + 54538 p^{6} T^{6} - 384 p^{9} T^{7} + p^{12} T^{8} \) |
| 37 | $D_4\times C_2$ | \( 1 + 536 T + 128278 T^{2} + 30933632 T^{3} + 8168593627 T^{4} + 30933632 p^{3} T^{5} + 128278 p^{6} T^{6} + 536 p^{9} T^{7} + p^{12} T^{8} \) |
| 41 | $D_{4}$ | \( ( 1 - 756 T + 278276 T^{2} - 756 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 43 | $D_{4}$ | \( ( 1 - 400 T + 142566 T^{2} - 400 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 47 | $D_4\times C_2$ | \( 1 + 312 T - 124838 T^{2} + 4535232 T^{3} + 28479079683 T^{4} + 4535232 p^{3} T^{5} - 124838 p^{6} T^{6} + 312 p^{9} T^{7} + p^{12} T^{8} \) |
| 53 | $D_4\times C_2$ | \( 1 + 52 T - 239278 T^{2} - 2900144 T^{3} + 35988363787 T^{4} - 2900144 p^{3} T^{5} - 239278 p^{6} T^{6} + 52 p^{9} T^{7} + p^{12} T^{8} \) |
| 59 | $D_4\times C_2$ | \( 1 + 864 T + 2534 p T^{2} + 160904448 T^{3} + 160901924475 T^{4} + 160904448 p^{3} T^{5} + 2534 p^{7} T^{6} + 864 p^{9} T^{7} + p^{12} T^{8} \) |
| 61 | $D_4\times C_2$ | \( 1 - 1416 T + 1066392 T^{2} - 686338032 T^{3} + 374460114599 T^{4} - 686338032 p^{3} T^{5} + 1066392 p^{6} T^{6} - 1416 p^{9} T^{7} + p^{12} T^{8} \) |
| 67 | $D_4\times C_2$ | \( 1 + 144 T - 529526 T^{2} - 7382016 T^{3} + 206093264907 T^{4} - 7382016 p^{3} T^{5} - 529526 p^{6} T^{6} + 144 p^{9} T^{7} + p^{12} T^{8} \) |
| 71 | $D_{4}$ | \( ( 1 - 1524 T + 1208266 T^{2} - 1524 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 73 | $D_4\times C_2$ | \( 1 - 744 T + 312240 T^{2} + 399333072 T^{3} - 308445380785 T^{4} + 399333072 p^{3} T^{5} + 312240 p^{6} T^{6} - 744 p^{9} T^{7} + p^{12} T^{8} \) |
| 79 | $D_4\times C_2$ | \( 1 + 976 T + 419842 T^{2} - 442463744 T^{3} - 428939059229 T^{4} - 442463744 p^{3} T^{5} + 419842 p^{6} T^{6} + 976 p^{9} T^{7} + p^{12} T^{8} \) |
| 83 | $D_{4}$ | \( ( 1 + 312 T - 644698 T^{2} + 312 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 89 | $D_4\times C_2$ | \( 1 + 108 T - 1318772 T^{2} - 8586216 T^{3} + 1264855900719 T^{4} - 8586216 p^{3} T^{5} - 1318772 p^{6} T^{6} + 108 p^{9} T^{7} + p^{12} T^{8} \) |
| 97 | $D_{4}$ | \( ( 1 - 744 T + 432480 T^{2} - 744 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
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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
−6.87632606527252600444542489130, −6.63435402554232289899969596618, −6.50994135632485623020398425925, −6.38979738854521593768568070889, −6.12489679631917629966206551459, −5.67161880275923541610645846802, −5.55600031177280125140090601833, −4.99971221706584919363118943696, −4.97781725839908971119899705546, −4.83971511523965663374739042027, −4.31607133646576587509335385882, −4.19243534058106496415949805305, −4.04116870444629167761492011433, −3.95542442351301626573021105019, −3.60468471299437802519719356598, −2.94699820172751735506954862945, −2.55018968910184049537498189922, −2.50852822176425141933881045826, −2.34432736295477954462089887187, −2.18269859558839755151234832493, −1.51788068164673556313036386908, −1.03569779480578654913395751210, −0.819270616589481615502111481695, −0.36357337352415531894280859600, −0.32167538334763363634013089493,
0.32167538334763363634013089493, 0.36357337352415531894280859600, 0.819270616589481615502111481695, 1.03569779480578654913395751210, 1.51788068164673556313036386908, 2.18269859558839755151234832493, 2.34432736295477954462089887187, 2.50852822176425141933881045826, 2.55018968910184049537498189922, 2.94699820172751735506954862945, 3.60468471299437802519719356598, 3.95542442351301626573021105019, 4.04116870444629167761492011433, 4.19243534058106496415949805305, 4.31607133646576587509335385882, 4.83971511523965663374739042027, 4.97781725839908971119899705546, 4.99971221706584919363118943696, 5.55600031177280125140090601833, 5.67161880275923541610645846802, 6.12489679631917629966206551459, 6.38979738854521593768568070889, 6.50994135632485623020398425925, 6.63435402554232289899969596618, 6.87632606527252600444542489130
