| 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 + 127-s + 131-s + 137-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 + 6.20e−5·127-s + 5.82e−5·131-s + 5.32e−5·137-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{48} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(5-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{48} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s+2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(0.3337148024\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3337148024\) |
| \(L(3)\) |
|
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 | \( ( 1 - 9788 T + 237997684 T^{2} - 2268092426756 T^{3} + 26342772579576982 T^{4} - 2268092426756 p^{4} T^{5} + 237997684 p^{8} T^{6} - 9788 p^{12} T^{7} + p^{16} T^{8} )^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{16} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−4.14707935264843975444553103442, −4.13930711093650479151915620385, −3.91533647873040451493770169005, −3.88381303009237298725432407335, −3.53345836380411451485856872489, −3.48481186961112226945938926102, −3.48275615673142125199941183808, −3.31426162051768117492103798778, −3.05281330737212868185291142097, −2.98757784130394356482375678091, −2.47049984589905954462042914532, −2.21050570693481058706932556870, −2.20777262534884561750757750797, −2.15262865370529601956389726503, −1.93837680190535841316456256580, −1.68250038076832970035320571102, −1.68169267599046545146189457467, −1.60071846366941736671275871138, −1.41083772136675680241573914903, −1.36568119532713669779005081966, −0.930561761765878013019885240281, −0.71470928557795258027414244536, −0.58028983157372590645960883776, −0.50609272099901067562483973631, −0.01915349785759697551152626574,
0.01915349785759697551152626574, 0.50609272099901067562483973631, 0.58028983157372590645960883776, 0.71470928557795258027414244536, 0.930561761765878013019885240281, 1.36568119532713669779005081966, 1.41083772136675680241573914903, 1.60071846366941736671275871138, 1.68169267599046545146189457467, 1.68250038076832970035320571102, 1.93837680190535841316456256580, 2.15262865370529601956389726503, 2.20777262534884561750757750797, 2.21050570693481058706932556870, 2.47049984589905954462042914532, 2.98757784130394356482375678091, 3.05281330737212868185291142097, 3.31426162051768117492103798778, 3.48275615673142125199941183808, 3.48481186961112226945938926102, 3.53345836380411451485856872489, 3.88381303009237298725432407335, 3.91533647873040451493770169005, 4.13930711093650479151915620385, 4.14707935264843975444553103442
Plot not available for L-functions of degree greater than 10.
