| L(s) = 1 | + 56·7-s + 200·9-s + 144·11-s + 1.20e3·23-s + 2.82e3·25-s + 1.39e3·29-s + 496·37-s − 5.87e3·43-s + 4.51e3·49-s − 1.68e3·53-s + 1.12e4·63-s + 1.33e4·67-s + 1.09e4·71-s + 8.06e3·77-s + 2.22e4·79-s + 2.40e4·81-s + 2.88e4·99-s − 1.39e3·107-s − 4.18e4·109-s − 7.70e4·113-s − 5.72e4·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + ⋯ |
| L(s) = 1 | + 8/7·7-s + 2.46·9-s + 1.19·11-s + 2.26·23-s + 4.51·25-s + 1.65·29-s + 0.362·37-s − 3.17·43-s + 1.88·49-s − 0.598·53-s + 2.82·63-s + 2.96·67-s + 2.18·71-s + 1.36·77-s + 3.56·79-s + 3.66·81-s + 2.93·99-s − 0.121·107-s − 3.52·109-s − 6.03·113-s − 3.91·121-s + 6.20e−5·127-s + 5.82e−5·131-s + 5.32e−5·137-s + 5.17e−5·139-s + 4.50e−5·149-s + 4.38e−5·151-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
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(88.49537682\) |
| \(L(\frac12)\) |
\(\approx\) |
\(88.49537682\) |
| \(L(3)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 - 8 p T - 1380 T^{2} + 968 p T^{3} + 23210 p^{3} T^{4} + 968 p^{5} T^{5} - 1380 p^{8} T^{6} - 8 p^{13} T^{7} + p^{16} T^{8} \) |
| good | 3 | \( 1 - 200 T^{2} + 15932 T^{4} - 50488 p^{2} T^{6} - 60506 p^{4} T^{8} - 50488 p^{10} T^{10} + 15932 p^{16} T^{12} - 200 p^{24} T^{14} + p^{32} T^{16} \) |
| 5 | \( 1 - 2824 T^{2} + 4042428 T^{4} - 3862105784 T^{6} + 2743366033286 T^{8} - 3862105784 p^{8} T^{10} + 4042428 p^{16} T^{12} - 2824 p^{24} T^{14} + p^{32} T^{16} \) |
| 11 | \( ( 1 - 72 T + 36412 T^{2} - 2210808 T^{3} + 748897990 T^{4} - 2210808 p^{4} T^{5} + 36412 p^{8} T^{6} - 72 p^{12} T^{7} + p^{16} T^{8} )^{2} \) |
| 13 | \( 1 - 48520 T^{2} + 2683023036 T^{4} - 87347075627576 T^{6} + 3257939383893547142 T^{8} - 87347075627576 p^{8} T^{10} + 2683023036 p^{16} T^{12} - 48520 p^{24} T^{14} + p^{32} T^{16} \) |
| 17 | \( 1 - 243720 T^{2} + 39301472796 T^{4} - 4688634560312376 T^{6} + \)\(45\!\cdots\!82\)\( T^{8} - 4688634560312376 p^{8} T^{10} + 39301472796 p^{16} T^{12} - 243720 p^{24} T^{14} + p^{32} T^{16} \) |
| 19 | \( 1 - 401224 T^{2} + 79830588732 T^{4} - 11505941347841144 T^{6} + \)\(14\!\cdots\!46\)\( T^{8} - 11505941347841144 p^{8} T^{10} + 79830588732 p^{16} T^{12} - 401224 p^{24} T^{14} + p^{32} T^{16} \) |
| 23 | \( ( 1 - 600 T + 846396 T^{2} - 378750696 T^{3} + 315664816262 T^{4} - 378750696 p^{4} T^{5} + 846396 p^{8} T^{6} - 600 p^{12} T^{7} + p^{16} T^{8} )^{2} \) |
| 29 | \( ( 1 - 24 p T + 2302300 T^{2} - 1369976712 T^{3} + 2273943916294 T^{4} - 1369976712 p^{4} T^{5} + 2302300 p^{8} T^{6} - 24 p^{13} T^{7} + p^{16} T^{8} )^{2} \) |
| 31 | \( 1 - 1851400 T^{2} + 3326880791580 T^{4} - 3646192948780345400 T^{6} + \)\(38\!\cdots\!62\)\( T^{8} - 3646192948780345400 p^{8} T^{10} + 3326880791580 p^{16} T^{12} - 1851400 p^{24} T^{14} + p^{32} T^{16} \) |
| 37 | \( ( 1 - 248 T + 4769884 T^{2} + 1362416056 T^{3} + 10105044932998 T^{4} + 1362416056 p^{4} T^{5} + 4769884 p^{8} T^{6} - 248 p^{12} T^{7} + p^{16} T^{8} )^{2} \) |
| 41 | \( 1 - 15990024 T^{2} + 116585968034076 T^{4} - \)\(52\!\cdots\!04\)\( T^{6} + \)\(17\!\cdots\!02\)\( T^{8} - \)\(52\!\cdots\!04\)\( p^{8} T^{10} + 116585968034076 p^{16} T^{12} - 15990024 p^{24} T^{14} + p^{32} T^{16} \) |
| 43 | \( ( 1 + 2936 T + 14422588 T^{2} + 29199968840 T^{3} + 75006402672838 T^{4} + 29199968840 p^{4} T^{5} + 14422588 p^{8} T^{6} + 2936 p^{12} T^{7} + p^{16} T^{8} )^{2} \) |
| 47 | \( 1 - 13319176 T^{2} + 128389654868508 T^{4} - \)\(87\!\cdots\!96\)\( T^{6} + \)\(49\!\cdots\!22\)\( T^{8} - \)\(87\!\cdots\!96\)\( p^{8} T^{10} + 128389654868508 p^{16} T^{12} - 13319176 p^{24} T^{14} + p^{32} T^{16} \) |
| 53 | \( ( 1 + 840 T + 11205724 T^{2} + 29618100600 T^{3} + 56705025448582 T^{4} + 29618100600 p^{4} T^{5} + 11205724 p^{8} T^{6} + 840 p^{12} T^{7} + p^{16} T^{8} )^{2} \) |
| 59 | \( 1 - 63583304 T^{2} + 1985998598470204 T^{4} - \)\(39\!\cdots\!84\)\( T^{6} + \)\(56\!\cdots\!50\)\( T^{8} - \)\(39\!\cdots\!84\)\( p^{8} T^{10} + 1985998598470204 p^{16} T^{12} - 63583304 p^{24} T^{14} + p^{32} T^{16} \) |
| 61 | \( 1 - 79555592 T^{2} + 3080299760506300 T^{4} - \)\(74\!\cdots\!92\)\( T^{6} + \)\(12\!\cdots\!62\)\( T^{8} - \)\(74\!\cdots\!92\)\( p^{8} T^{10} + 3080299760506300 p^{16} T^{12} - 79555592 p^{24} T^{14} + p^{32} T^{16} \) |
| 67 | \( ( 1 - 6664 T + 37471164 T^{2} - 223635767096 T^{3} + 1154752127076422 T^{4} - 223635767096 p^{4} T^{5} + 37471164 p^{8} T^{6} - 6664 p^{12} T^{7} + p^{16} T^{8} )^{2} \) |
| 71 | \( ( 1 - 5496 T + 78754076 T^{2} - 360814817352 T^{3} + 2689407667538118 T^{4} - 360814817352 p^{4} T^{5} + 78754076 p^{8} T^{6} - 5496 p^{12} T^{7} + p^{16} T^{8} )^{2} \) |
| 73 | \( 1 - 83180040 T^{2} + 5054619550429980 T^{4} - \)\(20\!\cdots\!60\)\( T^{6} + \)\(68\!\cdots\!54\)\( T^{8} - \)\(20\!\cdots\!60\)\( p^{8} T^{10} + 5054619550429980 p^{16} T^{12} - 83180040 p^{24} T^{14} + p^{32} T^{16} \) |
| 79 | \( ( 1 - 11128 T + 89885340 T^{2} - 599292860744 T^{3} + 3628496957560646 T^{4} - 599292860744 p^{4} T^{5} + 89885340 p^{8} T^{6} - 11128 p^{12} T^{7} + p^{16} T^{8} )^{2} \) |
| 83 | \( 1 - 168180168 T^{2} + 12468167299826748 T^{4} - \)\(60\!\cdots\!16\)\( T^{6} + \)\(26\!\cdots\!70\)\( T^{8} - \)\(60\!\cdots\!16\)\( p^{8} T^{10} + 12468167299826748 p^{16} T^{12} - 168180168 p^{24} T^{14} + p^{32} T^{16} \) |
| 89 | \( 1 - 162778632 T^{2} + 17574321098878236 T^{4} - \)\(11\!\cdots\!40\)\( T^{6} + \)\(79\!\cdots\!94\)\( T^{8} - \)\(11\!\cdots\!40\)\( p^{8} T^{10} + 17574321098878236 p^{16} T^{12} - 162778632 p^{24} T^{14} + p^{32} T^{16} \) |
| 97 | \( 1 - 410853384 T^{2} + 86521774961822748 T^{4} - \)\(12\!\cdots\!84\)\( T^{6} + \)\(12\!\cdots\!42\)\( T^{8} - \)\(12\!\cdots\!84\)\( p^{8} T^{10} + 86521774961822748 p^{16} T^{12} - 410853384 p^{24} T^{14} + p^{32} T^{16} \) |
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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
−4.08556526547082907271104843675, −4.00432674094352292999669904819, −3.93772637669761470521777103746, −3.81459376311432188367594013047, −3.78979682187088742619490058163, −3.54750805510914529093701898807, −3.18620202000879120278816364315, −3.13331938551280434314204080320, −3.03411531089114028605035374486, −2.82581440644995223059027312781, −2.67821987537128249892600698023, −2.65230892220385718178650826080, −2.37504789997833229872457063249, −2.05414823343355326889099162273, −1.84248137111750149362487782032, −1.80127860113689046386809135278, −1.73606408716822139068341852829, −1.31354173927184808611991465462, −1.18719962344489947568282388856, −1.14678537715436846092428573894, −0.950846000155142206605982625191, −0.880994271067756344407016206028, −0.68719730089540345425401717472, −0.47306349363318461905555736665, −0.31426085321143497437899445296,
0.31426085321143497437899445296, 0.47306349363318461905555736665, 0.68719730089540345425401717472, 0.880994271067756344407016206028, 0.950846000155142206605982625191, 1.14678537715436846092428573894, 1.18719962344489947568282388856, 1.31354173927184808611991465462, 1.73606408716822139068341852829, 1.80127860113689046386809135278, 1.84248137111750149362487782032, 2.05414823343355326889099162273, 2.37504789997833229872457063249, 2.65230892220385718178650826080, 2.67821987537128249892600698023, 2.82581440644995223059027312781, 3.03411531089114028605035374486, 3.13331938551280434314204080320, 3.18620202000879120278816364315, 3.54750805510914529093701898807, 3.78979682187088742619490058163, 3.81459376311432188367594013047, 3.93772637669761470521777103746, 4.00432674094352292999669904819, 4.08556526547082907271104843675
Plot not available for L-functions of degree greater than 10.
