| L(s) = 1 | − 36·5-s + 364·9-s − 236·13-s + 24·17-s − 1.54e3·25-s + 3.86e3·29-s − 2.40e3·37-s − 1.46e3·41-s − 1.31e4·45-s − 1.37e3·49-s + 6.96e3·53-s − 1.08e4·61-s + 8.49e3·65-s − 1.23e4·73-s + 7.35e4·81-s − 864·85-s − 1.62e4·89-s + 1.95e4·97-s − 5.99e4·101-s + 3.21e4·109-s + 2.83e4·113-s − 8.59e4·117-s + 3.44e4·121-s + 1.03e5·125-s + ⋯ |
| L(s) = 1 | − 1.43·5-s + 4.49·9-s − 1.39·13-s + 0.0830·17-s − 2.46·25-s + 4.59·29-s − 1.75·37-s − 0.870·41-s − 6.47·45-s − 4/7·49-s + 2.47·53-s − 2.91·61-s + 2.01·65-s − 2.31·73-s + 11.2·81-s − 0.119·85-s − 2.05·89-s + 2.08·97-s − 5.88·101-s + 2.70·109-s + 2.21·113-s − 6.27·117-s + 2.35·121-s + 6.62·125-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(5-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s+2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(4.044417254\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.044417254\) |
| \(L(3)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( ( 1 + p^{3} T^{2} )^{4} \) |
| good | 3 | \( 1 - 364 T^{2} + 58996 T^{4} - 665140 p^{2} T^{6} + 6128950 p^{4} T^{8} - 665140 p^{10} T^{10} + 58996 p^{16} T^{12} - 364 p^{24} T^{14} + p^{32} T^{16} \) |
| 5 | \( ( 1 + 18 T + 1256 T^{2} + 4374 T^{3} + 125238 p T^{4} + 4374 p^{4} T^{5} + 1256 p^{8} T^{6} + 18 p^{12} T^{7} + p^{16} T^{8} )^{2} \) |
| 11 | \( 1 - 34456 T^{2} + 684994300 T^{4} - 6757075983784 T^{6} + 75165791605017094 T^{8} - 6757075983784 p^{8} T^{10} + 684994300 p^{16} T^{12} - 34456 p^{24} T^{14} + p^{32} T^{16} \) |
| 13 | \( ( 1 + 118 T + 36112 T^{2} + 4638274 T^{3} + 395766430 T^{4} + 4638274 p^{4} T^{5} + 36112 p^{8} T^{6} + 118 p^{12} T^{7} + p^{16} T^{8} )^{2} \) |
| 17 | \( ( 1 - 12 T + 184628 T^{2} - 4685748 T^{3} + 20696679702 T^{4} - 4685748 p^{4} T^{5} + 184628 p^{8} T^{6} - 12 p^{12} T^{7} + p^{16} T^{8} )^{2} \) |
| 19 | \( 1 - 61036 T^{2} - 32814803084 T^{4} - 1630169398876 p T^{6} + \)\(82\!\cdots\!90\)\( T^{8} - 1630169398876 p^{9} T^{10} - 32814803084 p^{16} T^{12} - 61036 p^{24} T^{14} + p^{32} T^{16} \) |
| 23 | \( 1 - 412072 T^{2} + 5752684676 p T^{4} - 28500927574475416 T^{6} + \)\(96\!\cdots\!54\)\( T^{8} - 28500927574475416 p^{8} T^{10} + 5752684676 p^{17} T^{12} - 412072 p^{24} T^{14} + p^{32} T^{16} \) |
| 29 | \( ( 1 - 1932 T + 3041684 T^{2} - 2825631732 T^{3} + 2750375298198 T^{4} - 2825631732 p^{4} T^{5} + 3041684 p^{8} T^{6} - 1932 p^{12} T^{7} + p^{16} T^{8} )^{2} \) |
| 31 | \( 1 - 3534104 T^{2} + 6477716759164 T^{4} - 8287993744624356392 T^{6} + \)\(83\!\cdots\!86\)\( T^{8} - 8287993744624356392 p^{8} T^{10} + 6477716759164 p^{16} T^{12} - 3534104 p^{24} T^{14} + p^{32} T^{16} \) |
| 37 | \( ( 1 + 1204 T + 3257716 T^{2} - 13956404 T^{3} + 2706095859286 T^{4} - 13956404 p^{4} T^{5} + 3257716 p^{8} T^{6} + 1204 p^{12} T^{7} + p^{16} T^{8} )^{2} \) |
| 41 | \( ( 1 + 732 T + 2497988 T^{2} + 5124866532 T^{3} - 1351143982218 T^{4} + 5124866532 p^{4} T^{5} + 2497988 p^{8} T^{6} + 732 p^{12} T^{7} + p^{16} T^{8} )^{2} \) |
| 43 | \( 1 - 9448984 T^{2} + 53863850221564 T^{4} - \)\(25\!\cdots\!84\)\( T^{6} + \)\(99\!\cdots\!06\)\( T^{8} - \)\(25\!\cdots\!84\)\( p^{8} T^{10} + 53863850221564 p^{16} T^{12} - 9448984 p^{24} T^{14} + p^{32} T^{16} \) |
| 47 | \( 1 - 452008 p T^{2} + 221999395192060 T^{4} - \)\(16\!\cdots\!44\)\( T^{6} + \)\(90\!\cdots\!74\)\( T^{8} - \)\(16\!\cdots\!44\)\( p^{8} T^{10} + 221999395192060 p^{16} T^{12} - 452008 p^{25} T^{14} + p^{32} T^{16} \) |
| 53 | \( ( 1 - 3480 T + 20421500 T^{2} - 60170941992 T^{3} + 196509561519174 T^{4} - 60170941992 p^{4} T^{5} + 20421500 p^{8} T^{6} - 3480 p^{12} T^{7} + p^{16} T^{8} )^{2} \) |
| 59 | \( 1 - 42701932 T^{2} + 1100363717534452 T^{4} - \)\(20\!\cdots\!80\)\( T^{6} + \)\(27\!\cdots\!62\)\( T^{8} - \)\(20\!\cdots\!80\)\( p^{8} T^{10} + 1100363717534452 p^{16} T^{12} - 42701932 p^{24} T^{14} + p^{32} T^{16} \) |
| 61 | \( ( 1 + 5418 T + 27547384 T^{2} + 53238255966 T^{3} + 259141307924910 T^{4} + 53238255966 p^{4} T^{5} + 27547384 p^{8} T^{6} + 5418 p^{12} T^{7} + p^{16} T^{8} )^{2} \) |
| 67 | \( 1 - 114629672 T^{2} + 6227612495320156 T^{4} - \)\(21\!\cdots\!12\)\( T^{6} + \)\(50\!\cdots\!34\)\( T^{8} - \)\(21\!\cdots\!12\)\( p^{8} T^{10} + 6227612495320156 p^{16} T^{12} - 114629672 p^{24} T^{14} + p^{32} T^{16} \) |
| 71 | \( 1 - 91557832 T^{2} + 5208644673933724 T^{4} - \)\(19\!\cdots\!88\)\( T^{6} + \)\(57\!\cdots\!46\)\( T^{8} - \)\(19\!\cdots\!88\)\( p^{8} T^{10} + 5208644673933724 p^{16} T^{12} - 91557832 p^{24} T^{14} + p^{32} T^{16} \) |
| 73 | \( ( 1 + 6176 T + 90887404 T^{2} + 459954839840 T^{3} + 3596317489072678 T^{4} + 459954839840 p^{4} T^{5} + 90887404 p^{8} T^{6} + 6176 p^{12} T^{7} + p^{16} T^{8} )^{2} \) |
| 79 | \( 1 - 28491464 T^{2} + 4910319045660316 T^{4} - \)\(11\!\cdots\!40\)\( T^{6} + \)\(10\!\cdots\!70\)\( T^{8} - \)\(11\!\cdots\!40\)\( p^{8} T^{10} + 4910319045660316 p^{16} T^{12} - 28491464 p^{24} T^{14} + p^{32} T^{16} \) |
| 83 | \( 1 - 251706508 T^{2} + 32072905881661108 T^{4} - \)\(26\!\cdots\!64\)\( T^{6} + \)\(14\!\cdots\!46\)\( T^{8} - \)\(26\!\cdots\!64\)\( p^{8} T^{10} + 32072905881661108 p^{16} T^{12} - 251706508 p^{24} T^{14} + p^{32} T^{16} \) |
| 89 | \( ( 1 + 8136 T + 238896380 T^{2} + 1316646031608 T^{3} + 21713260826952198 T^{4} + 1316646031608 p^{4} T^{5} + 238896380 p^{8} T^{6} + 8136 p^{12} T^{7} + p^{16} T^{8} )^{2} \) |
| 97 | \( ( 1 - 9788 T + 237997684 T^{2} - 2268092426756 T^{3} + 26342772579576982 T^{4} - 2268092426756 p^{4} T^{5} + 237997684 p^{8} T^{6} - 9788 p^{12} T^{7} + p^{16} T^{8} )^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{16} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−5.63483464304786451761402996650, −5.02532370082343705109314311691, −4.99455657785465050773522267191, −4.89480965811893447690303321703, −4.88734853034560212540703613161, −4.48512551816049354555739552694, −4.38838609825162848602162881734, −4.34235933213961729328753658617, −4.17318593673444129606008345074, −4.08291658804494499147618437231, −3.64593892999049047160376400458, −3.60203457657536376819953849403, −3.34301926756322860235705931173, −3.28546079932441501360172349273, −2.83507021488450752018177520203, −2.55122830176367713256311980552, −2.35900143714427768655536510557, −2.11140163909709010462742278551, −1.81773569821903738771649775000, −1.48139320946916739215590598300, −1.37072429663287437397662562212, −1.23244321310849727161927281923, −0.854452002847232822156071670484, −0.32649713108637932214704635783, −0.30784787579071685226741452808,
0.30784787579071685226741452808, 0.32649713108637932214704635783, 0.854452002847232822156071670484, 1.23244321310849727161927281923, 1.37072429663287437397662562212, 1.48139320946916739215590598300, 1.81773569821903738771649775000, 2.11140163909709010462742278551, 2.35900143714427768655536510557, 2.55122830176367713256311980552, 2.83507021488450752018177520203, 3.28546079932441501360172349273, 3.34301926756322860235705931173, 3.60203457657536376819953849403, 3.64593892999049047160376400458, 4.08291658804494499147618437231, 4.17318593673444129606008345074, 4.34235933213961729328753658617, 4.38838609825162848602162881734, 4.48512551816049354555739552694, 4.88734853034560212540703613161, 4.89480965811893447690303321703, 4.99455657785465050773522267191, 5.02532370082343705109314311691, 5.63483464304786451761402996650
Plot not available for L-functions of degree greater than 10.
