| 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
| \(L(4)\) |
\(\approx\) |
\(0.01359601475\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.01359601475\) |
| \(L(\frac{9}{2})\) |
|
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 | $D_4\times C_2$ | \( 1 + 4098670 T + 11066174414449 T^{2} - \)\(63\!\cdots\!50\)\( T^{3} - \)\(78\!\cdots\!68\)\( T^{4} - \)\(63\!\cdots\!50\)\( p^{7} T^{5} + 11066174414449 p^{14} T^{6} + 4098670 p^{21} T^{7} + p^{28} T^{8} \) |
| show more | | |
| show less | | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.965690873889495235492535145981, −8.787122750312978573654394421954, −8.451383896729531522942746342209, −8.274948099258722391967896559118, −7.83630361185095359034289856860, −7.61348390936002952912783789080, −7.42057780900173510192859657890, −7.14647858863026412466020135547, −6.88156402191491428397361319870, −6.12718822591630372598392694069, −6.04961476349489908367202594276, −5.81153112250028029100647198888, −5.28146039116548533953762750929, −4.98908821232991632786516112457, −4.64865420775663160489088298364, −4.03251764576108916159645197968, −3.75037071180836264091732757065, −2.97376639608055154093305193977, −2.93900583060122940847668151462, −2.68322537880727914996201282083, −2.35653300627006348107225912953, −1.21805815557966426991407645997, −1.07639652773887397061009406295, −0.977647064736156044238460879109, −0.02152337680190510211394280358,
0.02152337680190510211394280358, 0.977647064736156044238460879109, 1.07639652773887397061009406295, 1.21805815557966426991407645997, 2.35653300627006348107225912953, 2.68322537880727914996201282083, 2.93900583060122940847668151462, 2.97376639608055154093305193977, 3.75037071180836264091732757065, 4.03251764576108916159645197968, 4.64865420775663160489088298364, 4.98908821232991632786516112457, 5.28146039116548533953762750929, 5.81153112250028029100647198888, 6.04961476349489908367202594276, 6.12718822591630372598392694069, 6.88156402191491428397361319870, 7.14647858863026412466020135547, 7.42057780900173510192859657890, 7.61348390936002952912783789080, 7.83630361185095359034289856860, 8.274948099258722391967896559118, 8.451383896729531522942746342209, 8.787122750312978573654394421954, 8.965690873889495235492535145981
