| L(s) = 1 | − 3·2-s + 4·4-s − 6·5-s + 39·8-s + 18·10-s − 6·11-s − 32·13-s − 125·16-s + 6·17-s + 64·19-s − 24·20-s + 18·22-s + 6·23-s + 202·25-s + 96·26-s + 504·29-s + 40·31-s + 252·32-s − 18·34-s + 248·37-s − 192·38-s − 234·40-s − 900·41-s + 752·43-s − 24·44-s − 18·46-s + 12·47-s + ⋯ |
| L(s) = 1 | − 1.06·2-s + 1/2·4-s − 0.536·5-s + 1.72·8-s + 0.569·10-s − 0.164·11-s − 0.682·13-s − 1.95·16-s + 0.0856·17-s + 0.772·19-s − 0.268·20-s + 0.174·22-s + 0.0543·23-s + 1.61·25-s + 0.724·26-s + 3.22·29-s + 0.231·31-s + 1.39·32-s − 0.0907·34-s + 1.10·37-s − 0.819·38-s − 0.924·40-s − 3.42·41-s + 2.66·43-s − 0.0822·44-s − 0.0576·46-s + 0.0372·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(2.133451801\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.133451801\) |
| \(L(\frac{5}{2})\) |
|
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 | | \( 1 \) |
| 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 | $D_{4}$ | \( ( 1 + 808 T + 903054 T^{2} + 808 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| show more | | |
| show less | | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.67040300192020027748427169652, −7.64135241977008896837526208929, −7.29737309779768208080400463177, −6.79434866212152922347428695626, −6.79292005794462095563738077326, −6.38140146775335437183001230643, −6.26672660076163992467146983176, −6.18576822211623830756651705686, −5.31383121792435877150821514093, −5.27997450529195642963789830790, −4.98141965968518431136609820929, −4.63618303678851795812543948322, −4.56835847267012128274196257061, −4.52653416079034541233277465235, −3.82306954167273412894877475754, −3.67886185264086729964300817287, −3.14946695947085764736892777683, −2.84666450946978389468064001712, −2.65673099837421436977593606398, −2.38946700980819698518632351688, −1.52423803825939629928149785508, −1.47285031319865108133670216426, −1.20381379075134864765827905941, −0.70217724294105440727984128171, −0.32053462587336742963830725481,
0.32053462587336742963830725481, 0.70217724294105440727984128171, 1.20381379075134864765827905941, 1.47285031319865108133670216426, 1.52423803825939629928149785508, 2.38946700980819698518632351688, 2.65673099837421436977593606398, 2.84666450946978389468064001712, 3.14946695947085764736892777683, 3.67886185264086729964300817287, 3.82306954167273412894877475754, 4.52653416079034541233277465235, 4.56835847267012128274196257061, 4.63618303678851795812543948322, 4.98141965968518431136609820929, 5.27997450529195642963789830790, 5.31383121792435877150821514093, 6.18576822211623830756651705686, 6.26672660076163992467146983176, 6.38140146775335437183001230643, 6.79292005794462095563738077326, 6.79434866212152922347428695626, 7.29737309779768208080400463177, 7.64135241977008896837526208929, 7.67040300192020027748427169652
