| L(s) = 1 | + 108·3-s − 672·5-s + 162·7-s + 7.29e3·9-s + 2.43e3·11-s − 5.66e3·13-s − 7.25e4·15-s − 7.73e3·17-s + 2.74e4·19-s + 1.74e4·21-s + 2.94e4·23-s + 8.39e4·25-s + 3.93e5·27-s + 1.31e5·29-s − 2.14e5·31-s + 2.63e5·33-s − 1.08e5·35-s − 3.24e5·37-s − 6.11e5·39-s − 1.22e6·41-s − 1.08e6·43-s − 4.89e6·45-s − 9.35e5·47-s − 2.56e6·49-s − 8.35e5·51-s − 3.98e5·53-s − 1.63e6·55-s + ⋯ |
| L(s) = 1 | + 2.30·3-s − 2.40·5-s + 0.178·7-s + 10/3·9-s + 0.551·11-s − 0.715·13-s − 5.55·15-s − 0.381·17-s + 0.917·19-s + 0.412·21-s + 0.505·23-s + 1.07·25-s + 3.84·27-s + 1.00·29-s − 1.29·31-s + 1.27·33-s − 0.429·35-s − 1.05·37-s − 1.65·39-s − 2.76·41-s − 2.07·43-s − 8.01·45-s − 1.31·47-s − 3.11·49-s − 0.881·51-s − 0.368·53-s − 1.32·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{4} \cdot 19^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(8-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{4} \cdot 19^{4}\right)^{s/2} \, \Gamma_{\C}(s+7/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(4)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{9}{2})\) |
|
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 | | \( 1 \) |
| 3 | $C_1$ | \( ( 1 - p^{3} T )^{4} \) |
| 19 | $C_1$ | \( ( 1 - p^{3} T )^{4} \) |
| good | 5 | $C_2 \wr S_4$ | \( 1 + 672 T + 367673 T^{2} + 1100826 p^{3} T^{3} + 1773040752 p^{2} T^{4} + 1100826 p^{10} T^{5} + 367673 p^{14} T^{6} + 672 p^{21} T^{7} + p^{28} T^{8} \) |
| 7 | $C_2 \wr S_4$ | \( 1 - 162 T + 2589341 T^{2} - 338196102 T^{3} + 415517565420 p T^{4} - 338196102 p^{7} T^{5} + 2589341 p^{14} T^{6} - 162 p^{21} T^{7} + p^{28} T^{8} \) |
| 11 | $C_2 \wr S_4$ | \( 1 - 2436 T + 60269161 T^{2} - 139425160350 T^{3} + 1595953556793140 T^{4} - 139425160350 p^{7} T^{5} + 60269161 p^{14} T^{6} - 2436 p^{21} T^{7} + p^{28} T^{8} \) |
| 13 | $C_2 \wr S_4$ | \( 1 + 5666 T + 133691888 T^{2} + 213592175574 T^{3} + 8377283699443694 T^{4} + 213592175574 p^{7} T^{5} + 133691888 p^{14} T^{6} + 5666 p^{21} T^{7} + p^{28} T^{8} \) |
| 17 | $C_2 \wr S_4$ | \( 1 + 7734 T + 943014841 T^{2} + 13004991383574 T^{3} + 446554180374703748 T^{4} + 13004991383574 p^{7} T^{5} + 943014841 p^{14} T^{6} + 7734 p^{21} T^{7} + p^{28} T^{8} \) |
| 23 | $C_2 \wr S_4$ | \( 1 - 29484 T + 3371967760 T^{2} + 142967581298052 T^{3} + 3804772659562392638 T^{4} + 142967581298052 p^{7} T^{5} + 3371967760 p^{14} T^{6} - 29484 p^{21} T^{7} + p^{28} T^{8} \) |
| 29 | $C_2 \wr S_4$ | \( 1 - 131910 T + 52797535832 T^{2} - 5788450790977410 T^{3} + \)\(12\!\cdots\!82\)\( T^{4} - 5788450790977410 p^{7} T^{5} + 52797535832 p^{14} T^{6} - 131910 p^{21} T^{7} + p^{28} T^{8} \) |
| 31 | $C_2 \wr S_4$ | \( 1 + 214346 T + 20662242152 T^{2} - 1481661696690918 T^{3} - \)\(54\!\cdots\!42\)\( T^{4} - 1481661696690918 p^{7} T^{5} + 20662242152 p^{14} T^{6} + 214346 p^{21} T^{7} + p^{28} T^{8} \) |
| 37 | $C_2 \wr S_4$ | \( 1 + 324066 T + 204212197028 T^{2} + 74815435371689838 T^{3} + \)\(27\!\cdots\!18\)\( T^{4} + 74815435371689838 p^{7} T^{5} + 204212197028 p^{14} T^{6} + 324066 p^{21} T^{7} + p^{28} T^{8} \) |
| 41 | $C_2 \wr S_4$ | \( 1 + 1220874 T + 917934986308 T^{2} + 440584776740323950 T^{3} + \)\(20\!\cdots\!34\)\( T^{4} + 440584776740323950 p^{7} T^{5} + 917934986308 p^{14} T^{6} + 1220874 p^{21} T^{7} + p^{28} T^{8} \) |
| 43 | $C_2 \wr S_4$ | \( 1 + 1080710 T + 1032172116341 T^{2} + 753935280844657770 T^{3} + \)\(42\!\cdots\!88\)\( T^{4} + 753935280844657770 p^{7} T^{5} + 1032172116341 p^{14} T^{6} + 1080710 p^{21} T^{7} + p^{28} T^{8} \) |
| 47 | $C_2 \wr S_4$ | \( 1 + 935946 T + 1498449936049 T^{2} + 924806897465938170 T^{3} + \)\(99\!\cdots\!52\)\( T^{4} + 924806897465938170 p^{7} T^{5} + 1498449936049 p^{14} T^{6} + 935946 p^{21} T^{7} + p^{28} T^{8} \) |
| 53 | $C_2 \wr S_4$ | \( 1 + 398946 T + 2217600413752 T^{2} + 881115161546099526 T^{3} + \)\(30\!\cdots\!14\)\( T^{4} + 881115161546099526 p^{7} T^{5} + 2217600413752 p^{14} T^{6} + 398946 p^{21} T^{7} + p^{28} T^{8} \) |
| 59 | $C_2 \wr S_4$ | \( 1 - 671580 T + 4250926437964 T^{2} + 1086839163963928068 T^{3} + \)\(96\!\cdots\!18\)\( T^{4} + 1086839163963928068 p^{7} T^{5} + 4250926437964 p^{14} T^{6} - 671580 p^{21} T^{7} + p^{28} T^{8} \) |
| 61 | $C_2 \wr S_4$ | \( 1 + 3506434 T + 11593629868609 T^{2} + 25374638692415740774 T^{3} + \)\(51\!\cdots\!84\)\( T^{4} + 25374638692415740774 p^{7} T^{5} + 11593629868609 p^{14} T^{6} + 3506434 p^{21} T^{7} + p^{28} T^{8} \) |
| 67 | $C_2 \wr S_4$ | \( 1 + 4541036 T + 11750835954476 T^{2} + 40678662931123001196 T^{3} + \)\(13\!\cdots\!58\)\( T^{4} + 40678662931123001196 p^{7} T^{5} + 11750835954476 p^{14} T^{6} + 4541036 p^{21} T^{7} + p^{28} T^{8} \) |
| 71 | $C_2 \wr S_4$ | \( 1 + 3864804 T + 20621546002156 T^{2} + 26998993483991422260 T^{3} + \)\(13\!\cdots\!30\)\( T^{4} + 26998993483991422260 p^{7} T^{5} + 20621546002156 p^{14} T^{6} + 3864804 p^{21} T^{7} + p^{28} T^{8} \) |
| 73 | $C_2 \wr S_4$ | \( 1 + 3133506 T + 44919684878873 T^{2} + 1384646054805594666 p T^{3} + \)\(74\!\cdots\!36\)\( T^{4} + 1384646054805594666 p^{8} T^{5} + 44919684878873 p^{14} T^{6} + 3133506 p^{21} T^{7} + p^{28} T^{8} \) |
| 79 | $C_2 \wr S_4$ | \( 1 + 6959610 T + 71697089167964 T^{2} + \)\(36\!\cdots\!70\)\( T^{3} + \)\(20\!\cdots\!22\)\( T^{4} + \)\(36\!\cdots\!70\)\( p^{7} T^{5} + 71697089167964 p^{14} T^{6} + 6959610 p^{21} T^{7} + p^{28} T^{8} \) |
| 83 | $C_2 \wr S_4$ | \( 1 + 8041302 T + 102730238536180 T^{2} + \)\(64\!\cdots\!58\)\( T^{3} + \)\(40\!\cdots\!66\)\( T^{4} + \)\(64\!\cdots\!58\)\( p^{7} T^{5} + 102730238536180 p^{14} T^{6} + 8041302 p^{21} T^{7} + p^{28} T^{8} \) |
| 89 | $C_2 \wr S_4$ | \( 1 + 1745418 T + 167087779811104 T^{2} + \)\(22\!\cdots\!66\)\( T^{3} + \)\(10\!\cdots\!50\)\( T^{4} + \)\(22\!\cdots\!66\)\( p^{7} T^{5} + 167087779811104 p^{14} T^{6} + 1745418 p^{21} T^{7} + p^{28} T^{8} \) |
| 97 | $C_2 \wr S_4$ | \( 1 + 15738304 T + 281713263458956 T^{2} + \)\(32\!\cdots\!32\)\( T^{3} + \)\(33\!\cdots\!58\)\( T^{4} + \)\(32\!\cdots\!32\)\( p^{7} T^{5} + 281713263458956 p^{14} T^{6} + 15738304 p^{21} T^{7} + p^{28} T^{8} \) |
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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
−8.275178696781651657789871418736, −7.73168256345546906766572995104, −7.59869759668439651979717209739, −7.55492409598229331743404621572, −7.47947073733461091105138115347, −6.86905122715702758980951451164, −6.61150004457404900281218667145, −6.46814003945258966129163923047, −6.43820847638400466373441379714, −5.32864198886111925389875256443, −5.26071328240617879141785941859, −5.08575488259073423480627609128, −4.77143459368577776113178788420, −4.19707611052074538448151625127, −4.03907680500928504075651342918, −3.91684776858197598723781322015, −3.76155788371403663621696131202, −3.03868131652148029392613623411, −3.03601163841238888216538121385, −2.97299070153659330759071886608, −2.67602222644694211011622639077, −1.71671327634309690405199103458, −1.51686225565182522447085752689, −1.45941676980200749253428584507, −1.44006143780200296821467240539, 0, 0, 0, 0,
1.44006143780200296821467240539, 1.45941676980200749253428584507, 1.51686225565182522447085752689, 1.71671327634309690405199103458, 2.67602222644694211011622639077, 2.97299070153659330759071886608, 3.03601163841238888216538121385, 3.03868131652148029392613623411, 3.76155788371403663621696131202, 3.91684776858197598723781322015, 4.03907680500928504075651342918, 4.19707611052074538448151625127, 4.77143459368577776113178788420, 5.08575488259073423480627609128, 5.26071328240617879141785941859, 5.32864198886111925389875256443, 6.43820847638400466373441379714, 6.46814003945258966129163923047, 6.61150004457404900281218667145, 6.86905122715702758980951451164, 7.47947073733461091105138115347, 7.55492409598229331743404621572, 7.59869759668439651979717209739, 7.73168256345546906766572995104, 8.275178696781651657789871418736
