| L(s) = 1 | + 9·2-s + 130·4-s + 180·5-s − 700·7-s + 477·8-s + 1.62e3·10-s + 1.08e4·11-s + 5.48e3·13-s − 6.30e3·14-s + 1.03e4·16-s − 3.28e4·17-s + 3.20e4·19-s + 2.34e4·20-s + 9.80e4·22-s + 2.43e4·23-s + 1.63e5·25-s + 4.93e4·26-s − 9.10e4·28-s − 1.43e5·29-s + 3.87e4·31-s − 8.77e4·32-s − 2.95e5·34-s − 1.26e5·35-s + 9.11e5·37-s + 2.88e5·38-s + 8.58e4·40-s + 7.31e5·41-s + ⋯ |
| L(s) = 1 | + 0.795·2-s + 1.01·4-s + 0.643·5-s − 0.771·7-s + 0.329·8-s + 0.512·10-s + 2.46·11-s + 0.691·13-s − 0.613·14-s + 0.629·16-s − 1.62·17-s + 1.07·19-s + 0.654·20-s + 1.96·22-s + 0.417·23-s + 2.09·25-s + 0.550·26-s − 0.783·28-s − 1.09·29-s + 0.233·31-s − 0.473·32-s − 1.28·34-s − 0.496·35-s + 2.95·37-s + 0.852·38-s + 0.212·40-s + 1.65·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 43046721 ^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43046721 ^{s/2} \, \Gamma_{\C}(s+7/2)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(15.33167037\) |
| \(L(\frac12)\) |
\(\approx\) |
\(15.33167037\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | | \( 1 \) |
| good | 2 | $D_4\times C_2$ | \( 1 - 9 T - 49 T^{2} + 567 p T^{3} - 2463 p^{2} T^{4} + 567 p^{8} T^{5} - 49 p^{14} T^{6} - 9 p^{21} T^{7} + p^{28} T^{8} \) |
| 5 | $D_4\times C_2$ | \( 1 - 36 p T - 26273 p T^{2} - 54108 p^{2} T^{3} + 727368384 p^{2} T^{4} - 54108 p^{9} T^{5} - 26273 p^{15} T^{6} - 36 p^{22} T^{7} + p^{28} T^{8} \) |
| 7 | $D_4\times C_2$ | \( 1 + 100 p T - 94961 T^{2} - 106212500 p T^{3} - 739579771328 T^{4} - 106212500 p^{8} T^{5} - 94961 p^{14} T^{6} + 100 p^{22} T^{7} + p^{28} T^{8} \) |
| 11 | $D_4\times C_2$ | \( 1 - 90 p^{2} T + 50203733 T^{2} - 2647262250 p^{2} T^{3} + 1986194375394348 T^{4} - 2647262250 p^{9} T^{5} + 50203733 p^{14} T^{6} - 90 p^{23} T^{7} + p^{28} T^{8} \) |
| 13 | $D_4\times C_2$ | \( 1 - 5480 T - 27158234 T^{2} + 374330032000 T^{3} - 2551199591830133 T^{4} + 374330032000 p^{7} T^{5} - 27158234 p^{14} T^{6} - 5480 p^{21} T^{7} + p^{28} T^{8} \) |
| 17 | $D_{4}$ | \( ( 1 + 16416 T + 79007650 T^{2} + 16416 p^{7} T^{3} + p^{14} T^{4} )^{2} \) |
| 19 | $D_{4}$ | \( ( 1 - 16024 T + 1645526562 T^{2} - 16024 p^{7} T^{3} + p^{14} T^{4} )^{2} \) |
| 23 | $D_4\times C_2$ | \( 1 - 24372 T - 2311612066 T^{2} + 95149371189168 T^{3} - 5172734676275436957 T^{4} + 95149371189168 p^{7} T^{5} - 2311612066 p^{14} T^{6} - 24372 p^{21} T^{7} + p^{28} T^{8} \) |
| 29 | $D_4\times C_2$ | \( 1 + 143280 T - 19092997318 T^{2} + 733937916168000 T^{3} + \)\(91\!\cdots\!43\)\( T^{4} + 733937916168000 p^{7} T^{5} - 19092997318 p^{14} T^{6} + 143280 p^{21} T^{7} + p^{28} T^{8} \) |
| 31 | $D_4\times C_2$ | \( 1 - 38708 T - 24998494289 T^{2} + 1104278262087652 T^{3} - 96085741293891320816 T^{4} + 1104278262087652 p^{7} T^{5} - 24998494289 p^{14} T^{6} - 38708 p^{21} T^{7} + p^{28} T^{8} \) |
| 37 | $D_{4}$ | \( ( 1 - 455620 T + 173526750366 T^{2} - 455620 p^{7} T^{3} + p^{14} T^{4} )^{2} \) |
| 41 | $D_4\times C_2$ | \( 1 - 731880 T + 1529648818 p T^{2} - 61056492590988000 T^{3} + \)\(81\!\cdots\!83\)\( T^{4} - 61056492590988000 p^{7} T^{5} + 1529648818 p^{15} T^{6} - 731880 p^{21} T^{7} + p^{28} T^{8} \) |
| 43 | $D_4\times C_2$ | \( 1 - 1088840 T + 434479093486 T^{2} - 225886641364316000 T^{3} + \)\(16\!\cdots\!47\)\( T^{4} - 225886641364316000 p^{7} T^{5} + 434479093486 p^{14} T^{6} - 1088840 p^{21} T^{7} + p^{28} T^{8} \) |
| 47 | $D_4\times C_2$ | \( 1 - 1561500 T + 1014040055534 T^{2} - 641770181452710000 T^{3} + \)\(53\!\cdots\!87\)\( T^{4} - 641770181452710000 p^{7} T^{5} + 1014040055534 p^{14} T^{6} - 1561500 p^{21} T^{7} + p^{28} T^{8} \) |
| 53 | $D_{4}$ | \( ( 1 + 2610468 T + 3933113324245 T^{2} + 2610468 p^{7} T^{3} + p^{14} T^{4} )^{2} \) |
| 59 | $D_4\times C_2$ | \( 1 - 1731960 T - 1329825064438 T^{2} + 1121950635256656000 T^{3} + \)\(50\!\cdots\!83\)\( T^{4} + 1121950635256656000 p^{7} T^{5} - 1329825064438 p^{14} T^{6} - 1731960 p^{21} T^{7} + p^{28} T^{8} \) |
| 61 | $D_4\times C_2$ | \( 1 - 620192 T - 5918398022954 T^{2} - 10884659710932992 T^{3} + \)\(28\!\cdots\!39\)\( T^{4} - 10884659710932992 p^{7} T^{5} - 5918398022954 p^{14} T^{6} - 620192 p^{21} T^{7} + p^{28} T^{8} \) |
| 67 | $D_4\times C_2$ | \( 1 + 346600 T - 12016990578146 T^{2} + 5441248271500000 T^{3} + \)\(10\!\cdots\!87\)\( T^{4} + 5441248271500000 p^{7} T^{5} - 12016990578146 p^{14} T^{6} + 346600 p^{21} T^{7} + p^{28} T^{8} \) |
| 71 | $D_{4}$ | \( ( 1 - 4242240 T + 14648438075182 T^{2} - 4242240 p^{7} T^{3} + p^{14} T^{4} )^{2} \) |
| 73 | $D_{4}$ | \( ( 1 + 3145190 T + 18855097518219 T^{2} + 3145190 p^{7} T^{3} + p^{14} T^{4} )^{2} \) |
| 79 | $D_4\times C_2$ | \( 1 + 10110616 T + 44597210011234 T^{2} + \)\(19\!\cdots\!64\)\( T^{3} + \)\(98\!\cdots\!99\)\( T^{4} + \)\(19\!\cdots\!64\)\( p^{7} T^{5} + 44597210011234 p^{14} T^{6} + 10110616 p^{21} T^{7} + p^{28} T^{8} \) |
| 83 | $D_4\times C_2$ | \( 1 - 644202 T - 52162543092091 T^{2} + 1091640661370608518 T^{3} + \)\(20\!\cdots\!88\)\( T^{4} + 1091640661370608518 p^{7} T^{5} - 52162543092091 p^{14} T^{6} - 644202 p^{21} T^{7} + p^{28} T^{8} \) |
| 89 | $D_{4}$ | \( ( 1 + 6021000 T + 94843763242558 T^{2} + 6021000 p^{7} T^{3} + p^{14} T^{4} )^{2} \) |
| 97 | $D_4\times C_2$ | \( 1 + 4098670 T + 11066174414449 T^{2} - \)\(63\!\cdots\!50\)\( T^{3} - \)\(78\!\cdots\!68\)\( T^{4} - \)\(63\!\cdots\!50\)\( p^{7} T^{5} + 11066174414449 p^{14} T^{6} + 4098670 p^{21} T^{7} + p^{28} T^{8} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.147000163318842966502704820346, −9.085976549780191183495623436939, −8.717442245752585103139702473012, −8.130469170762493019243330713547, −7.995041841250873261047577050087, −7.23726401240407476905593892751, −7.07389507520542995188969958167, −7.05021905278371807614594404258, −6.54344262097206123906175491819, −6.08264685069164416558688674615, −6.02199072790136231819718268545, −5.93035627402637886080380259787, −5.45345918739039282150495541183, −4.70924464777523734253800048001, −4.53213403965099846041649382477, −4.15626497689373217558523284511, −3.67675125820217519040409566759, −3.60513770763791984470470695147, −2.80394686992945066770321098049, −2.62233453003962693047894731134, −2.44337210471920643119098780893, −1.45160080976710607710921842289, −1.35729216525320073169786058208, −0.948979629451854245493220394137, −0.47277622131960460452056500387,
0.47277622131960460452056500387, 0.948979629451854245493220394137, 1.35729216525320073169786058208, 1.45160080976710607710921842289, 2.44337210471920643119098780893, 2.62233453003962693047894731134, 2.80394686992945066770321098049, 3.60513770763791984470470695147, 3.67675125820217519040409566759, 4.15626497689373217558523284511, 4.53213403965099846041649382477, 4.70924464777523734253800048001, 5.45345918739039282150495541183, 5.93035627402637886080380259787, 6.02199072790136231819718268545, 6.08264685069164416558688674615, 6.54344262097206123906175491819, 7.05021905278371807614594404258, 7.07389507520542995188969958167, 7.23726401240407476905593892751, 7.995041841250873261047577050087, 8.130469170762493019243330713547, 8.717442245752585103139702473012, 9.085976549780191183495623436939, 9.147000163318842966502704820346
