| L(s) = 1 | + 3·2-s − 6·3-s + 4·4-s − 6·5-s − 18·6-s − 39·8-s + 9·9-s − 18·10-s + 6·11-s − 24·12-s + 32·13-s + 36·15-s − 125·16-s + 6·17-s + 27·18-s − 64·19-s − 24·20-s + 18·22-s − 6·23-s + 234·24-s + 202·25-s + 96·26-s + 54·27-s − 504·29-s + 108·30-s − 40·31-s − 252·32-s + ⋯ |
| L(s) = 1 | + 1.06·2-s − 1.15·3-s + 1/2·4-s − 0.536·5-s − 1.22·6-s − 1.72·8-s + 1/3·9-s − 0.569·10-s + 0.164·11-s − 0.577·12-s + 0.682·13-s + 0.619·15-s − 1.95·16-s + 0.0856·17-s + 0.353·18-s − 0.772·19-s − 0.268·20-s + 0.174·22-s − 0.0543·23-s + 1.99·24-s + 1.61·25-s + 0.724·26-s + 0.384·27-s − 3.22·29-s + 0.657·30-s − 0.231·31-s − 1.39·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{4} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{4} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.849329802\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.849329802\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | $C_2$ | \( ( 1 + p T + p^{2} T^{2} )^{2} \) |
| 7 | | \( 1 \) |
| good | 2 | $D_4\times C_2$ | \( 1 - 3 T + 5 T^{2} + 9 p^{2} T^{3} - 15 p^{3} T^{4} + 9 p^{5} T^{5} + 5 p^{6} T^{6} - 3 p^{9} T^{7} + p^{12} T^{8} \) |
| 5 | $D_4\times C_2$ | \( 1 + 6 T - 166 T^{2} - 288 T^{3} + 20679 T^{4} - 288 p^{3} T^{5} - 166 p^{6} T^{6} + 6 p^{9} T^{7} + p^{12} T^{8} \) |
| 11 | $D_4\times C_2$ | \( 1 - 6 T - 10 p^{2} T^{2} + 8496 T^{3} - 266961 T^{4} + 8496 p^{3} T^{5} - 10 p^{8} T^{6} - 6 p^{9} T^{7} + p^{12} T^{8} \) |
| 13 | $D_{4}$ | \( ( 1 - 16 T + 2406 T^{2} - 16 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 17 | $D_4\times C_2$ | \( 1 - 6 T - 9742 T^{2} + 288 T^{3} + 71294847 T^{4} + 288 p^{3} T^{5} - 9742 p^{6} T^{6} - 6 p^{9} T^{7} + p^{12} T^{8} \) |
| 19 | $D_4\times C_2$ | \( 1 + 64 T - 2438 T^{2} - 459776 T^{3} - 32447189 T^{4} - 459776 p^{3} T^{5} - 2438 p^{6} T^{6} + 64 p^{9} T^{7} + p^{12} T^{8} \) |
| 23 | $D_4\times C_2$ | \( 1 + 6 T - 7834 T^{2} - 98784 T^{3} - 86537001 T^{4} - 98784 p^{3} T^{5} - 7834 p^{6} T^{6} + 6 p^{9} T^{7} + p^{12} T^{8} \) |
| 29 | $D_{4}$ | \( ( 1 + 252 T + 56446 T^{2} + 252 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 31 | $D_4\times C_2$ | \( 1 + 40 T + 15490 T^{2} - 2938880 T^{3} - 742237181 T^{4} - 2938880 p^{3} T^{5} + 15490 p^{6} T^{6} + 40 p^{9} T^{7} + p^{12} T^{8} \) |
| 37 | $D_4\times C_2$ | \( 1 - 248 T - 36710 T^{2} + 766816 T^{3} + 3964901275 T^{4} + 766816 p^{3} T^{5} - 36710 p^{6} T^{6} - 248 p^{9} T^{7} + p^{12} T^{8} \) |
| 41 | $D_{4}$ | \( ( 1 + 450 T + 175642 T^{2} + 450 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 43 | $D_{4}$ | \( ( 1 - 376 T + 161526 T^{2} - 376 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 47 | $D_4\times C_2$ | \( 1 - 12 T - 141646 T^{2} + 790272 T^{3} + 9310238259 T^{4} + 790272 p^{3} T^{5} - 141646 p^{6} T^{6} - 12 p^{9} T^{7} + p^{12} T^{8} \) |
| 53 | $D_4\times C_2$ | \( 1 - 1104 T + 616586 T^{2} - 336141504 T^{3} + 159062942139 T^{4} - 336141504 p^{3} T^{5} + 616586 p^{6} T^{6} - 1104 p^{9} T^{7} + p^{12} T^{8} \) |
| 59 | $D_4\times C_2$ | \( 1 + 804 T + 265802 T^{2} - 24235776 T^{3} - 30073788309 T^{4} - 24235776 p^{3} T^{5} + 265802 p^{6} T^{6} + 804 p^{9} T^{7} + p^{12} T^{8} \) |
| 61 | $D_4\times C_2$ | \( 1 - 428 T - 242702 T^{2} + 12016528 T^{3} + 88279223131 T^{4} + 12016528 p^{3} T^{5} - 242702 p^{6} T^{6} - 428 p^{9} T^{7} + p^{12} T^{8} \) |
| 67 | $D_4\times C_2$ | \( 1 + 148 T - 418886 T^{2} - 23788928 T^{3} + 97249529179 T^{4} - 23788928 p^{3} T^{5} - 418886 p^{6} T^{6} + 148 p^{9} T^{7} + p^{12} T^{8} \) |
| 71 | $D_{4}$ | \( ( 1 - 954 T + 13106 p T^{2} - 954 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 73 | $D_4\times C_2$ | \( 1 + 1072 T + 85906 T^{2} + 305781568 T^{3} + 532173766867 T^{4} + 305781568 p^{3} T^{5} + 85906 p^{6} T^{6} + 1072 p^{9} T^{7} + p^{12} T^{8} \) |
| 79 | $D_4\times C_2$ | \( 1 - 572 T - 574478 T^{2} + 48285952 T^{3} + 408592434547 T^{4} + 48285952 p^{3} T^{5} - 574478 p^{6} T^{6} - 572 p^{9} T^{7} + p^{12} T^{8} \) |
| 83 | $D_{4}$ | \( ( 1 - 1944 T + 1957030 T^{2} - 1944 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 89 | $D_4\times C_2$ | \( 1 + 366 T - 1022134 T^{2} - 92908368 T^{3} + 745127969775 T^{4} - 92908368 p^{3} T^{5} - 1022134 p^{6} T^{6} + 366 p^{9} T^{7} + p^{12} T^{8} \) |
| 97 | $D_{4}$ | \( ( 1 - 808 T + 903054 T^{2} - 808 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
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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
−9.177907353008261095770822983213, −8.760719017446982162370182245723, −8.546333973051242560177809555081, −8.541444901186451992417462657827, −7.75325424186489321260587495585, −7.56656589518672363562460124831, −7.48338794200151657963292108352, −6.82775709170678877849820223506, −6.57400547805011098065439205085, −6.52530525755590830632894217856, −6.02841152472127532824674950929, −5.85245536681459585870757937992, −5.61399278249297958231216988588, −5.26264740013150256294273396789, −4.87754027020575796590093327537, −4.83334181100551577078033078384, −4.02414526520975542504704162540, −3.75157181735387410894784269478, −3.58241608040808237628814492452, −3.37567009777173051964010573087, −2.54313998316529335130109602507, −2.36401543535289497793551425317, −1.66913228257120325637199557086, −0.66922079192933098214161963031, −0.43905095468108892615897437985,
0.43905095468108892615897437985, 0.66922079192933098214161963031, 1.66913228257120325637199557086, 2.36401543535289497793551425317, 2.54313998316529335130109602507, 3.37567009777173051964010573087, 3.58241608040808237628814492452, 3.75157181735387410894784269478, 4.02414526520975542504704162540, 4.83334181100551577078033078384, 4.87754027020575796590093327537, 5.26264740013150256294273396789, 5.61399278249297958231216988588, 5.85245536681459585870757937992, 6.02841152472127532824674950929, 6.52530525755590830632894217856, 6.57400547805011098065439205085, 6.82775709170678877849820223506, 7.48338794200151657963292108352, 7.56656589518672363562460124831, 7.75325424186489321260587495585, 8.541444901186451992417462657827, 8.546333973051242560177809555081, 8.760719017446982162370182245723, 9.177907353008261095770822983213
