| L(s) = 1 | + 79·4-s − 324·9-s + 372·11-s + 3.74e3·16-s + 5.48e3·19-s − 1.14e3·29-s − 1.88e4·31-s − 2.55e4·36-s − 3.40e4·41-s + 2.93e4·44-s − 9.60e3·49-s − 1.95e4·59-s − 1.20e5·61-s + 1.28e5·64-s + 2.02e3·71-s + 4.33e5·76-s − 2.98e5·79-s + 6.56e4·81-s + 1.59e4·89-s − 1.20e5·99-s − 5.94e5·101-s + 2.57e5·109-s − 9.00e4·116-s − 7.01e5·121-s − 1.48e6·124-s + 127-s + 131-s + ⋯ |
| L(s) = 1 | + 2.46·4-s − 4/3·9-s + 0.926·11-s + 3.65·16-s + 3.48·19-s − 0.251·29-s − 3.51·31-s − 3.29·36-s − 3.15·41-s + 2.28·44-s − 4/7·49-s − 0.731·59-s − 4.15·61-s + 3.91·64-s + 0.0477·71-s + 8.60·76-s − 5.37·79-s + 10/9·81-s + 0.213·89-s − 1.23·99-s − 5.80·101-s + 2.07·109-s − 0.621·116-s − 4.35·121-s − 8.68·124-s + 5.50e−6·127-s + 5.09e−6·131-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 5^{16} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(6-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 5^{16} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s+5/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(6.828471526\) |
| \(L(\frac12)\) |
\(\approx\) |
\(6.828471526\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( ( 1 + p^{4} T^{2} )^{4} \) |
| 5 | \( 1 \) |
| 7 | \( ( 1 + p^{4} T^{2} )^{4} \) |
| good | 2 | \( 1 - 79 T^{2} + 39 p^{6} T^{4} - 461 p^{6} T^{6} - 391 p^{6} T^{8} - 461 p^{16} T^{10} + 39 p^{26} T^{12} - 79 p^{30} T^{14} + p^{40} T^{16} \) |
| 11 | \( ( 1 - 186 T + 402761 T^{2} + 2659314 T^{3} + 68817190120 T^{4} + 2659314 p^{5} T^{5} + 402761 p^{10} T^{6} - 186 p^{15} T^{7} + p^{20} T^{8} )^{2} \) |
| 13 | \( 1 - 1302914 T^{2} + 1014559689753 T^{4} - 529225396328384998 T^{6} + \)\(22\!\cdots\!92\)\( T^{8} - 529225396328384998 p^{10} T^{10} + 1014559689753 p^{20} T^{12} - 1302914 p^{30} T^{14} + p^{40} T^{16} \) |
| 17 | \( 1 - 9605650 T^{2} + 42096041652969 T^{4} - \)\(11\!\cdots\!70\)\( T^{6} + \)\(19\!\cdots\!60\)\( T^{8} - \)\(11\!\cdots\!70\)\( p^{10} T^{10} + 42096041652969 p^{20} T^{12} - 9605650 p^{30} T^{14} + p^{40} T^{16} \) |
| 19 | \( ( 1 - 2742 T + 9912864 T^{2} - 17174987862 T^{3} + 36523787688110 T^{4} - 17174987862 p^{5} T^{5} + 9912864 p^{10} T^{6} - 2742 p^{15} T^{7} + p^{20} T^{8} )^{2} \) |
| 23 | \( 1 - 32689468 T^{2} + 538796313439794 T^{4} - \)\(57\!\cdots\!28\)\( T^{6} + \)\(43\!\cdots\!75\)\( T^{8} - \)\(57\!\cdots\!28\)\( p^{10} T^{10} + 538796313439794 p^{20} T^{12} - 32689468 p^{30} T^{14} + p^{40} T^{16} \) |
| 29 | \( ( 1 + 570 T + 55924020 T^{2} - 8689553436 T^{3} + 1441178925934253 T^{4} - 8689553436 p^{5} T^{5} + 55924020 p^{10} T^{6} + 570 p^{15} T^{7} + p^{20} T^{8} )^{2} \) |
| 31 | \( ( 1 + 9414 T + 65308645 T^{2} + 184313662974 T^{3} + 919886646484032 T^{4} + 184313662974 p^{5} T^{5} + 65308645 p^{10} T^{6} + 9414 p^{15} T^{7} + p^{20} T^{8} )^{2} \) |
| 37 | \( 1 - 22203926 T^{2} + 8450361395665993 T^{4} - \)\(20\!\cdots\!22\)\( T^{6} + \)\(63\!\cdots\!12\)\( T^{8} - \)\(20\!\cdots\!22\)\( p^{10} T^{10} + 8450361395665993 p^{20} T^{12} - 22203926 p^{30} T^{14} + p^{40} T^{16} \) |
| 41 | \( ( 1 + 17004 T + 425964287 T^{2} + 4662047157972 T^{3} + 68836655275245772 T^{4} + 4662047157972 p^{5} T^{5} + 425964287 p^{10} T^{6} + 17004 p^{15} T^{7} + p^{20} T^{8} )^{2} \) |
| 43 | \( 1 - 888424844 T^{2} + 376459025216386578 T^{4} - \)\(98\!\cdots\!08\)\( T^{6} + \)\(17\!\cdots\!67\)\( T^{8} - \)\(98\!\cdots\!08\)\( p^{10} T^{10} + 376459025216386578 p^{20} T^{12} - 888424844 p^{30} T^{14} + p^{40} T^{16} \) |
| 47 | \( 1 - 1092435804 T^{2} + 488306643429367252 T^{4} - \)\(12\!\cdots\!08\)\( T^{6} + \)\(25\!\cdots\!94\)\( T^{8} - \)\(12\!\cdots\!08\)\( p^{10} T^{10} + 488306643429367252 p^{20} T^{12} - 1092435804 p^{30} T^{14} + p^{40} T^{16} \) |
| 53 | \( 1 - 1956734542 T^{2} + 1908544482620353473 T^{4} - \)\(12\!\cdots\!10\)\( T^{6} + \)\(61\!\cdots\!92\)\( T^{8} - \)\(12\!\cdots\!10\)\( p^{10} T^{10} + 1908544482620353473 p^{20} T^{12} - 1956734542 p^{30} T^{14} + p^{40} T^{16} \) |
| 59 | \( ( 1 + 9780 T + 1569939921 T^{2} + 27486034055814 T^{3} + 1219412476221851096 T^{4} + 27486034055814 p^{5} T^{5} + 1569939921 p^{10} T^{6} + 9780 p^{15} T^{7} + p^{20} T^{8} )^{2} \) |
| 61 | \( ( 1 + 60408 T + 4675983199 T^{2} + 164501404686972 T^{3} + 6504972629954951940 T^{4} + 164501404686972 p^{5} T^{5} + 4675983199 p^{10} T^{6} + 60408 p^{15} T^{7} + p^{20} T^{8} )^{2} \) |
| 67 | \( 1 - 2128153862 T^{2} + 1230335109360249369 T^{4} - \)\(36\!\cdots\!62\)\( T^{6} + \)\(92\!\cdots\!40\)\( T^{8} - \)\(36\!\cdots\!62\)\( p^{10} T^{10} + 1230335109360249369 p^{20} T^{12} - 2128153862 p^{30} T^{14} + p^{40} T^{16} \) |
| 71 | \( ( 1 - 1014 T + 1307507589 T^{2} - 49493354960046 T^{3} + 1894193992085195540 T^{4} - 49493354960046 p^{5} T^{5} + 1307507589 p^{10} T^{6} - 1014 p^{15} T^{7} + p^{20} T^{8} )^{2} \) |
| 73 | \( 1 - 1868958500 T^{2} + 13418161801111977540 T^{4} - \)\(18\!\cdots\!76\)\( T^{6} + \)\(76\!\cdots\!38\)\( T^{8} - \)\(18\!\cdots\!76\)\( p^{10} T^{10} + 13418161801111977540 p^{20} T^{12} - 1868958500 p^{30} T^{14} + p^{40} T^{16} \) |
| 79 | \( ( 1 + 149160 T + 16094243151 T^{2} + 1159429909717092 T^{3} + 72337890216436558556 T^{4} + 1159429909717092 p^{5} T^{5} + 16094243151 p^{10} T^{6} + 149160 p^{15} T^{7} + p^{20} T^{8} )^{2} \) |
| 83 | \( 1 - 856445286 T^{2} + 12861796470088235617 T^{4} - \)\(59\!\cdots\!02\)\( T^{6} + \)\(72\!\cdots\!64\)\( T^{8} - \)\(59\!\cdots\!02\)\( p^{10} T^{10} + 12861796470088235617 p^{20} T^{12} - 856445286 p^{30} T^{14} + p^{40} T^{16} \) |
| 89 | \( ( 1 - 7962 T + 19461670324 T^{2} - 124884626825526 T^{3} + \)\(15\!\cdots\!70\)\( T^{4} - 124884626825526 p^{5} T^{5} + 19461670324 p^{10} T^{6} - 7962 p^{15} T^{7} + p^{20} T^{8} )^{2} \) |
| 97 | \( 1 - 45012727896 T^{2} + \)\(10\!\cdots\!08\)\( T^{4} - \)\(15\!\cdots\!32\)\( T^{6} + \)\(15\!\cdots\!82\)\( T^{8} - \)\(15\!\cdots\!32\)\( p^{10} T^{10} + \)\(10\!\cdots\!08\)\( p^{20} T^{12} - 45012727896 p^{30} T^{14} + p^{40} T^{16} \) |
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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
−3.72916651025742857953116442402, −3.69477942760246546393659325973, −3.55216388939719498317561435746, −3.35676992694534939884000657347, −3.25827025167749894964131722805, −3.24522606035943850415438793014, −2.95754503984303266987849391151, −2.85648949273743909416498225058, −2.80173860235002391624262228690, −2.58925446570479111881741485995, −2.58216937714481567065289918654, −2.31208028015445244166891406283, −2.19544021623455173963755361159, −1.69494616753253742749552019030, −1.63706549497140200123825643321, −1.57875726260710994970242581411, −1.49555622783450661680113572562, −1.46319605723644883870114278915, −1.35237066485135011035717462833, −1.32896116750793419546143829976, −0.803861535285439850263471702932, −0.52316293243293751890867984694, −0.39108050637573729328894735508, −0.33944150571486145953012140117, −0.11682145747655642540741354530,
0.11682145747655642540741354530, 0.33944150571486145953012140117, 0.39108050637573729328894735508, 0.52316293243293751890867984694, 0.803861535285439850263471702932, 1.32896116750793419546143829976, 1.35237066485135011035717462833, 1.46319605723644883870114278915, 1.49555622783450661680113572562, 1.57875726260710994970242581411, 1.63706549497140200123825643321, 1.69494616753253742749552019030, 2.19544021623455173963755361159, 2.31208028015445244166891406283, 2.58216937714481567065289918654, 2.58925446570479111881741485995, 2.80173860235002391624262228690, 2.85648949273743909416498225058, 2.95754503984303266987849391151, 3.24522606035943850415438793014, 3.25827025167749894964131722805, 3.35676992694534939884000657347, 3.55216388939719498317561435746, 3.69477942760246546393659325973, 3.72916651025742857953116442402
Plot not available for L-functions of degree greater than 10.
