| L(s) = 1 | + 3·2-s + 7·4-s − 6·5-s + 3·8-s − 18·10-s + 46·13-s − 5·16-s + 6·17-s − 42·20-s − 67·25-s + 138·26-s − 42·29-s − 45·32-s + 18·34-s + 28·37-s − 18·40-s − 84·41-s + 167·49-s − 201·50-s + 322·52-s − 36·53-s − 126·58-s + 34·61-s − 85·64-s − 276·65-s + 42·68-s + 58·73-s + ⋯ |
| L(s) = 1 | + 3/2·2-s + 7/4·4-s − 6/5·5-s + 3/8·8-s − 9/5·10-s + 3.53·13-s − 0.312·16-s + 6/17·17-s − 2.09·20-s − 2.67·25-s + 5.30·26-s − 1.44·29-s − 1.40·32-s + 9/17·34-s + 0.756·37-s − 0.449·40-s − 2.04·41-s + 3.40·49-s − 4.01·50-s + 6.19·52-s − 0.679·53-s − 2.17·58-s + 0.557·61-s − 1.32·64-s − 4.24·65-s + 0.617·68-s + 0.794·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{32}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{32}\right)^{s/2} \, \Gamma_{\C}(s+1)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.4790738273\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4790738273\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - 3 T + p T^{2} + 3 p^{2} T^{3} - 9 p^{2} T^{4} + 3 p^{4} T^{5} + p^{5} T^{6} - 3 p^{6} T^{7} + p^{8} T^{8} \) |
| 3 | \( 1 \) |
| good | 5 | \( ( 1 + 3 T + 47 T^{2} - 6 p T^{3} + 804 T^{4} - 6 p^{3} T^{5} + 47 p^{4} T^{6} + 3 p^{6} T^{7} + p^{8} T^{8} )^{2} \) |
| 7 | \( 1 - 167 T^{2} + 14701 T^{4} - 959570 T^{6} + 51275386 T^{8} - 959570 p^{4} T^{10} + 14701 p^{8} T^{12} - 167 p^{12} T^{14} + p^{16} T^{16} \) |
| 11 | \( 1 - 524 T^{2} + 134482 T^{4} - 22990832 T^{6} + 3065351611 T^{8} - 22990832 p^{4} T^{10} + 134482 p^{8} T^{12} - 524 p^{12} T^{14} + p^{16} T^{16} \) |
| 13 | \( ( 1 - 23 T + 805 T^{2} - 11762 T^{3} + 214858 T^{4} - 11762 p^{2} T^{5} + 805 p^{4} T^{6} - 23 p^{6} T^{7} + p^{8} T^{8} )^{2} \) |
| 17 | \( ( 1 - 3 T + 2 p^{2} T^{2} - 6549 T^{3} + 169242 T^{4} - 6549 p^{2} T^{5} + 2 p^{6} T^{6} - 3 p^{6} T^{7} + p^{8} T^{8} )^{2} \) |
| 19 | \( 1 - 1673 T^{2} + 1559890 T^{4} - 937716023 T^{6} + 401371470970 T^{8} - 937716023 p^{4} T^{10} + 1559890 p^{8} T^{12} - 1673 p^{12} T^{14} + p^{16} T^{16} \) |
| 23 | \( 1 - 2687 T^{2} + 3500317 T^{4} - 2971858562 T^{6} + 1824468134410 T^{8} - 2971858562 p^{4} T^{10} + 3500317 p^{8} T^{12} - 2687 p^{12} T^{14} + p^{16} T^{16} \) |
| 29 | \( ( 1 + 21 T + 2873 T^{2} + 39750 T^{3} + 3357342 T^{4} + 39750 p^{2} T^{5} + 2873 p^{4} T^{6} + 21 p^{6} T^{7} + p^{8} T^{8} )^{2} \) |
| 31 | \( 1 - 4715 T^{2} + 11501641 T^{4} - 18322192394 T^{6} + 20727776670694 T^{8} - 18322192394 p^{4} T^{10} + 11501641 p^{8} T^{12} - 4715 p^{12} T^{14} + p^{16} T^{16} \) |
| 37 | \( ( 1 - 14 T + 3040 T^{2} - 66050 T^{3} + 123814 p T^{4} - 66050 p^{2} T^{5} + 3040 p^{4} T^{6} - 14 p^{6} T^{7} + p^{8} T^{8} )^{2} \) |
| 41 | \( ( 1 + 42 T + 6188 T^{2} + 192468 T^{3} + 15263853 T^{4} + 192468 p^{2} T^{5} + 6188 p^{4} T^{6} + 42 p^{6} T^{7} + p^{8} T^{8} )^{2} \) |
| 43 | \( 1 - 10292 T^{2} + 53105962 T^{4} - 172233596384 T^{6} + 381742267864051 T^{8} - 172233596384 p^{4} T^{10} + 53105962 p^{8} T^{12} - 10292 p^{12} T^{14} + p^{16} T^{16} \) |
| 47 | \( 1 - 12983 T^{2} + 78335149 T^{4} - 294042673394 T^{6} + 766851321282202 T^{8} - 294042673394 p^{4} T^{10} + 78335149 p^{8} T^{12} - 12983 p^{12} T^{14} + p^{16} T^{16} \) |
| 53 | \( ( 1 + 18 T + 10016 T^{2} + 126558 T^{3} + 40472766 T^{4} + 126558 p^{2} T^{5} + 10016 p^{4} T^{6} + 18 p^{6} T^{7} + p^{8} T^{8} )^{2} \) |
| 59 | \( 1 - 18092 T^{2} + 151416802 T^{4} - 803198812400 T^{6} + 3155407101414283 T^{8} - 803198812400 p^{4} T^{10} + 151416802 p^{8} T^{12} - 18092 p^{12} T^{14} + p^{16} T^{16} \) |
| 61 | \( ( 1 - 17 T + 3871 T^{2} - 95678 T^{3} + 31663708 T^{4} - 95678 p^{2} T^{5} + 3871 p^{4} T^{6} - 17 p^{6} T^{7} + p^{8} T^{8} )^{2} \) |
| 67 | \( 1 - 25148 T^{2} + 302080258 T^{4} - 2289407522192 T^{6} + 12118574906350123 T^{8} - 2289407522192 p^{4} T^{10} + 302080258 p^{8} T^{12} - 25148 p^{12} T^{14} + p^{16} T^{16} \) |
| 71 | \( 1 - 12968 T^{2} + 137492380 T^{4} - 912038324888 T^{6} + 5454839368725190 T^{8} - 912038324888 p^{4} T^{10} + 137492380 p^{8} T^{12} - 12968 p^{12} T^{14} + p^{16} T^{16} \) |
| 73 | \( ( 1 - 29 T + 7774 T^{2} - 620747 T^{3} + 46170850 T^{4} - 620747 p^{2} T^{5} + 7774 p^{4} T^{6} - 29 p^{6} T^{7} + p^{8} T^{8} )^{2} \) |
| 79 | \( 1 - 293 p T^{2} + 207724873 T^{4} - 836021205386 T^{6} + 2483172469984678 T^{8} - 836021205386 p^{4} T^{10} + 207724873 p^{8} T^{12} - 293 p^{13} T^{14} + p^{16} T^{16} \) |
| 83 | \( 1 - 17795 T^{2} + 288209353 T^{4} - 2684318937722 T^{6} + 22778145105251062 T^{8} - 2684318937722 p^{4} T^{10} + 288209353 p^{8} T^{12} - 17795 p^{12} T^{14} + p^{16} T^{16} \) |
| 89 | \( ( 1 + 96 T + 27716 T^{2} + 1940016 T^{3} + 308634966 T^{4} + 1940016 p^{2} T^{5} + 27716 p^{4} T^{6} + 96 p^{6} T^{7} + p^{8} T^{8} )^{2} \) |
| 97 | \( ( 1 - 74 T + 35344 T^{2} - 2077892 T^{3} + 488109505 T^{4} - 2077892 p^{2} T^{5} + 35344 p^{4} T^{6} - 74 p^{6} T^{7} + p^{8} T^{8} )^{2} \) |
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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
−4.97374109173128524611572906204, −4.73747546398372825344894007487, −4.51141646506301676950787096393, −4.42664485140482178386009563521, −4.32176700822171224470145320251, −3.90537198580977447128822548668, −3.90142136721426197374423654344, −3.85271010372455451105295257947, −3.80392860387399370354517765240, −3.52851487323532191580642910567, −3.47890443752757833834977683247, −3.30802202263756542240535372543, −3.30162795069335999858116750070, −2.94369608194720709518345101877, −2.77917679961882609003043094855, −2.44580538296053995822145154045, −2.27371969722570352120463837926, −2.04076512479219093908813302710, −2.03804463753001354282295997573, −1.66167348176210732916213929602, −1.32194231028093260926574485004, −1.31440648896698477934391180333, −0.793034492184583340968955691538, −0.64591862760490925403838855967, −0.05414555848176728866269865512,
0.05414555848176728866269865512, 0.64591862760490925403838855967, 0.793034492184583340968955691538, 1.31440648896698477934391180333, 1.32194231028093260926574485004, 1.66167348176210732916213929602, 2.03804463753001354282295997573, 2.04076512479219093908813302710, 2.27371969722570352120463837926, 2.44580538296053995822145154045, 2.77917679961882609003043094855, 2.94369608194720709518345101877, 3.30162795069335999858116750070, 3.30802202263756542240535372543, 3.47890443752757833834977683247, 3.52851487323532191580642910567, 3.80392860387399370354517765240, 3.85271010372455451105295257947, 3.90142136721426197374423654344, 3.90537198580977447128822548668, 4.32176700822171224470145320251, 4.42664485140482178386009563521, 4.51141646506301676950787096393, 4.73747546398372825344894007487, 4.97374109173128524611572906204
Plot not available for L-functions of degree greater than 10.
