| L(s) = 1 | − 2·2-s − 5·4-s − 20·5-s + 4·7-s + 10·8-s + 40·10-s − 10·11-s + 52·13-s − 8·14-s + 3·16-s − 212·17-s − 46·19-s + 100·20-s + 20·22-s − 104·23-s + 250·25-s − 104·26-s − 20·28-s − 202·29-s − 60·31-s − 34·32-s + 424·34-s − 80·35-s + 802·37-s + 92·38-s − 200·40-s + 258·41-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 5/8·4-s − 1.78·5-s + 0.215·7-s + 0.441·8-s + 1.26·10-s − 0.274·11-s + 1.10·13-s − 0.152·14-s + 3/64·16-s − 3.02·17-s − 0.555·19-s + 1.11·20-s + 0.193·22-s − 0.942·23-s + 2·25-s − 0.784·26-s − 0.134·28-s − 1.29·29-s − 0.347·31-s − 0.187·32-s + 2.13·34-s − 0.386·35-s + 3.56·37-s + 0.392·38-s − 0.790·40-s + 0.982·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 5^{4} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 5^{4} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.136928669\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.136928669\) |
| \(L(\frac{5}{2})\) |
|
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 | | \( 1 \) |
| 5 | $C_1$ | \( ( 1 + p T )^{4} \) |
| 13 | $C_1$ | \( ( 1 - p T )^{4} \) |
| good | 2 | $C_2 \wr S_4$ | \( 1 + p T + 9 T^{2} + 9 p T^{3} + 29 p T^{4} + 9 p^{4} T^{5} + 9 p^{6} T^{6} + p^{10} T^{7} + p^{12} T^{8} \) |
| 7 | $C_2 \wr S_4$ | \( 1 - 4 T + 377 T^{2} + 702 p T^{3} + 110060 T^{4} + 702 p^{4} T^{5} + 377 p^{6} T^{6} - 4 p^{9} T^{7} + p^{12} T^{8} \) |
| 11 | $C_2 \wr S_4$ | \( 1 + 10 T + 2397 T^{2} + 61902 T^{3} + 3478876 T^{4} + 61902 p^{3} T^{5} + 2397 p^{6} T^{6} + 10 p^{9} T^{7} + p^{12} T^{8} \) |
| 17 | $C_2 \wr S_4$ | \( 1 + 212 T + 33441 T^{2} + 3350694 T^{3} + 277528240 T^{4} + 3350694 p^{3} T^{5} + 33441 p^{6} T^{6} + 212 p^{9} T^{7} + p^{12} T^{8} \) |
| 19 | $C_2 \wr S_4$ | \( 1 + 46 T + 400 p T^{2} + 176278 T^{3} + 62905678 T^{4} + 176278 p^{3} T^{5} + 400 p^{7} T^{6} + 46 p^{9} T^{7} + p^{12} T^{8} \) |
| 23 | $C_2 \wr S_4$ | \( 1 + 104 T + 29769 T^{2} + 1401090 T^{3} + 385189948 T^{4} + 1401090 p^{3} T^{5} + 29769 p^{6} T^{6} + 104 p^{9} T^{7} + p^{12} T^{8} \) |
| 29 | $C_2 \wr S_4$ | \( 1 + 202 T + 74304 T^{2} + 11666478 T^{3} + 2665137550 T^{4} + 11666478 p^{3} T^{5} + 74304 p^{6} T^{6} + 202 p^{9} T^{7} + p^{12} T^{8} \) |
| 31 | $C_2 \wr S_4$ | \( 1 + 60 T + 78512 T^{2} + 7494156 T^{3} + 2874703134 T^{4} + 7494156 p^{3} T^{5} + 78512 p^{6} T^{6} + 60 p^{9} T^{7} + p^{12} T^{8} \) |
| 37 | $C_2 \wr S_4$ | \( 1 - 802 T + 402473 T^{2} - 137046990 T^{3} + 35851939652 T^{4} - 137046990 p^{3} T^{5} + 402473 p^{6} T^{6} - 802 p^{9} T^{7} + p^{12} T^{8} \) |
| 41 | $C_2 \wr S_4$ | \( 1 - 258 T + 275873 T^{2} - 51028542 T^{3} + 28560421068 T^{4} - 51028542 p^{3} T^{5} + 275873 p^{6} T^{6} - 258 p^{9} T^{7} + p^{12} T^{8} \) |
| 43 | $C_2 \wr S_4$ | \( 1 - 824 T + 350380 T^{2} - 98396408 T^{3} + 26366200246 T^{4} - 98396408 p^{3} T^{5} + 350380 p^{6} T^{6} - 824 p^{9} T^{7} + p^{12} T^{8} \) |
| 47 | $C_2 \wr S_4$ | \( 1 - 476 T + 281232 T^{2} - 98781516 T^{3} + 33442219294 T^{4} - 98781516 p^{3} T^{5} + 281232 p^{6} T^{6} - 476 p^{9} T^{7} + p^{12} T^{8} \) |
| 53 | $C_2 \wr S_4$ | \( 1 + 66 T + 414449 T^{2} + 33100614 T^{3} + 79167814884 T^{4} + 33100614 p^{3} T^{5} + 414449 p^{6} T^{6} + 66 p^{9} T^{7} + p^{12} T^{8} \) |
| 59 | $C_2 \wr S_4$ | \( 1 - 1164 T + 1105772 T^{2} - 658415340 T^{3} + 354020917110 T^{4} - 658415340 p^{3} T^{5} + 1105772 p^{6} T^{6} - 1164 p^{9} T^{7} + p^{12} T^{8} \) |
| 61 | $C_2 \wr S_4$ | \( 1 + 858 T + 522569 T^{2} + 333572514 T^{3} + 194113150284 T^{4} + 333572514 p^{3} T^{5} + 522569 p^{6} T^{6} + 858 p^{9} T^{7} + p^{12} T^{8} \) |
| 67 | $C_2 \wr S_4$ | \( 1 + 448 T + 1139788 T^{2} + 388934080 T^{3} + 506344537174 T^{4} + 388934080 p^{3} T^{5} + 1139788 p^{6} T^{6} + 448 p^{9} T^{7} + p^{12} T^{8} \) |
| 71 | $C_2 \wr S_4$ | \( 1 + 414 T + 533453 T^{2} + 191299014 T^{3} + 215370415284 T^{4} + 191299014 p^{3} T^{5} + 533453 p^{6} T^{6} + 414 p^{9} T^{7} + p^{12} T^{8} \) |
| 73 | $C_2 \wr S_4$ | \( 1 - 778 T + 1007000 T^{2} - 542121606 T^{3} + 456185126510 T^{4} - 542121606 p^{3} T^{5} + 1007000 p^{6} T^{6} - 778 p^{9} T^{7} + p^{12} T^{8} \) |
| 79 | $C_2 \wr S_4$ | \( 1 + 438 T + 1807493 T^{2} + 545475486 T^{3} + 1283789000724 T^{4} + 545475486 p^{3} T^{5} + 1807493 p^{6} T^{6} + 438 p^{9} T^{7} + p^{12} T^{8} \) |
| 83 | $C_2 \wr S_4$ | \( 1 + 416 T + 1057884 T^{2} - 50135904 T^{3} + 504741837046 T^{4} - 50135904 p^{3} T^{5} + 1057884 p^{6} T^{6} + 416 p^{9} T^{7} + p^{12} T^{8} \) |
| 89 | $C_2 \wr S_4$ | \( 1 + 526 T + 2816937 T^{2} + 1093296342 T^{3} + 2978978439172 T^{4} + 1093296342 p^{3} T^{5} + 2816937 p^{6} T^{6} + 526 p^{9} T^{7} + p^{12} T^{8} \) |
| 97 | $C_2 \wr S_4$ | \( 1 - 1540 T + 3360881 T^{2} - 3534072258 T^{3} + 4653597355688 T^{4} - 3534072258 p^{3} T^{5} + 3360881 p^{6} T^{6} - 1540 p^{9} T^{7} + p^{12} T^{8} \) |
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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
−7.42184707477461350288385364165, −7.10926698592490578088979394079, −7.09133984536620383997214322226, −6.40607853935177808522954427610, −6.36360465208116815592498939334, −6.17848505510505931441942427944, −5.96753496701168922592460618373, −5.54446985628602446781446629467, −5.49996849622640845352414716204, −4.74122876938705142281840215371, −4.69229709811701502198799141808, −4.49283811152327265391070815737, −4.20030636099574501119887079671, −3.96279326547131458350159903377, −3.96024161288914160554145053952, −3.69256127991114542850608696260, −3.03478033163425779794831806272, −2.69509649299370518466010378672, −2.60636037164518770235358780156, −2.02484298154665116552401725448, −1.95213492925844039833276347491, −1.26319415833524365904581392234, −0.70956891350769545527647115404, −0.45362568536389969842417152395, −0.39996283955807847701873988939,
0.39996283955807847701873988939, 0.45362568536389969842417152395, 0.70956891350769545527647115404, 1.26319415833524365904581392234, 1.95213492925844039833276347491, 2.02484298154665116552401725448, 2.60636037164518770235358780156, 2.69509649299370518466010378672, 3.03478033163425779794831806272, 3.69256127991114542850608696260, 3.96024161288914160554145053952, 3.96279326547131458350159903377, 4.20030636099574501119887079671, 4.49283811152327265391070815737, 4.69229709811701502198799141808, 4.74122876938705142281840215371, 5.49996849622640845352414716204, 5.54446985628602446781446629467, 5.96753496701168922592460618373, 6.17848505510505931441942427944, 6.36360465208116815592498939334, 6.40607853935177808522954427610, 7.09133984536620383997214322226, 7.10926698592490578088979394079, 7.42184707477461350288385364165
