| L(s) = 1 | − 32·2-s − 53·3-s + 640·4-s + 111·5-s + 1.69e3·6-s − 1.66e3·7-s − 1.02e4·8-s − 665·9-s − 3.55e3·10-s − 4.59e3·11-s − 3.39e4·12-s + 7.84e3·13-s + 5.33e4·14-s − 5.88e3·15-s + 1.43e5·16-s + 2.31e4·17-s + 2.12e4·18-s + 2.36e4·19-s + 7.10e4·20-s + 8.82e4·21-s + 1.46e5·22-s + 2.41e4·23-s + 5.42e5·24-s − 2.19e5·25-s − 2.51e5·26-s − 2.10e4·27-s − 1.06e6·28-s + ⋯ |
| L(s) = 1 | − 2.82·2-s − 1.13·3-s + 5·4-s + 0.397·5-s + 3.20·6-s − 1.83·7-s − 7.07·8-s − 0.304·9-s − 1.12·10-s − 1.04·11-s − 5.66·12-s + 0.990·13-s + 5.19·14-s − 0.450·15-s + 35/4·16-s + 1.14·17-s + 0.860·18-s + 0.792·19-s + 1.98·20-s + 2.08·21-s + 2.94·22-s + 0.413·23-s + 8.01·24-s − 2.80·25-s − 2.80·26-s − 0.205·27-s − 9.17·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 29986576 ^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 29986576 ^{s/2} \, \Gamma_{\C}(s+7/2)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | $C_1$ | \( ( 1 + p^{3} T )^{4} \) |
| 37 | $C_1$ | \( ( 1 - p^{3} T )^{4} \) |
| good | 3 | $C_2 \wr S_4$ | \( 1 + 53 T + 386 p^{2} T^{2} + 2968 p^{4} T^{3} + 348569 p^{3} T^{4} + 2968 p^{11} T^{5} + 386 p^{16} T^{6} + 53 p^{21} T^{7} + p^{28} T^{8} \) |
| 5 | $C_2 \wr S_4$ | \( 1 - 111 T + 46297 p T^{2} - 1302973 p^{2} T^{3} + 190908864 p^{3} T^{4} - 1302973 p^{9} T^{5} + 46297 p^{15} T^{6} - 111 p^{21} T^{7} + p^{28} T^{8} \) |
| 7 | $C_2 \wr S_4$ | \( 1 + 34 p^{2} T + 3836251 T^{2} + 3980351580 T^{3} + 4954084913556 T^{4} + 3980351580 p^{7} T^{5} + 3836251 p^{14} T^{6} + 34 p^{23} T^{7} + p^{28} T^{8} \) |
| 11 | $C_2 \wr S_4$ | \( 1 + 4593 T + 58690784 T^{2} + 139324724144 T^{3} + 1358111660813121 T^{4} + 139324724144 p^{7} T^{5} + 58690784 p^{14} T^{6} + 4593 p^{21} T^{7} + p^{28} T^{8} \) |
| 13 | $C_2 \wr S_4$ | \( 1 - 7847 T + 44905309 T^{2} + 238876335675 T^{3} - 5523621661124304 T^{4} + 238876335675 p^{7} T^{5} + 44905309 p^{14} T^{6} - 7847 p^{21} T^{7} + p^{28} T^{8} \) |
| 17 | $C_2 \wr S_4$ | \( 1 - 23172 T + 1565149604 T^{2} - 26309358229308 T^{3} + 956973477754784886 T^{4} - 26309358229308 p^{7} T^{5} + 1565149604 p^{14} T^{6} - 23172 p^{21} T^{7} + p^{28} T^{8} \) |
| 19 | $C_2 \wr S_4$ | \( 1 - 23696 T + 2348541316 T^{2} - 44830471203152 T^{3} + 2917224547808407606 T^{4} - 44830471203152 p^{7} T^{5} + 2348541316 p^{14} T^{6} - 23696 p^{21} T^{7} + p^{28} T^{8} \) |
| 23 | $C_2 \wr S_4$ | \( 1 - 24105 T + 7013385317 T^{2} - 88359223748697 T^{3} + 29217779771576109072 T^{4} - 88359223748697 p^{7} T^{5} + 7013385317 p^{14} T^{6} - 24105 p^{21} T^{7} + p^{28} T^{8} \) |
| 29 | $C_2 \wr S_4$ | \( 1 + 140949 T + 69012666881 T^{2} + 6785207860407915 T^{3} + \)\(17\!\cdots\!36\)\( T^{4} + 6785207860407915 p^{7} T^{5} + 69012666881 p^{14} T^{6} + 140949 p^{21} T^{7} + p^{28} T^{8} \) |
| 31 | $C_2 \wr S_4$ | \( 1 + 21439 p T + 256765050331 T^{2} + 67005695462988077 T^{3} + \)\(12\!\cdots\!88\)\( T^{4} + 67005695462988077 p^{7} T^{5} + 256765050331 p^{14} T^{6} + 21439 p^{22} T^{7} + p^{28} T^{8} \) |
| 41 | $C_2 \wr S_4$ | \( 1 + 709737 T + 268358061962 T^{2} + 179477185301479320 T^{3} + \)\(11\!\cdots\!17\)\( T^{4} + 179477185301479320 p^{7} T^{5} + 268358061962 p^{14} T^{6} + 709737 p^{21} T^{7} + p^{28} T^{8} \) |
| 43 | $C_2 \wr S_4$ | \( 1 + 128962 T + 680217086800 T^{2} + 72403133514238178 T^{3} + \)\(24\!\cdots\!46\)\( T^{4} + 72403133514238178 p^{7} T^{5} + 680217086800 p^{14} T^{6} + 128962 p^{21} T^{7} + p^{28} T^{8} \) |
| 47 | $C_2 \wr S_4$ | \( 1 + 9486 p T + 1135342318835 T^{2} + 614279906235197180 T^{3} + \)\(71\!\cdots\!84\)\( T^{4} + 614279906235197180 p^{7} T^{5} + 1135342318835 p^{14} T^{6} + 9486 p^{22} T^{7} + p^{28} T^{8} \) |
| 53 | $C_2 \wr S_4$ | \( 1 + 975870 T + 2112432693629 T^{2} - 517582713481480554 T^{3} + \)\(79\!\cdots\!80\)\( T^{4} - 517582713481480554 p^{7} T^{5} + 2112432693629 p^{14} T^{6} + 975870 p^{21} T^{7} + p^{28} T^{8} \) |
| 59 | $C_2 \wr S_4$ | \( 1 + 1812858 T + 9776352637904 T^{2} + 11765178797880874346 T^{3} + \)\(35\!\cdots\!58\)\( T^{4} + 11765178797880874346 p^{7} T^{5} + 9776352637904 p^{14} T^{6} + 1812858 p^{21} T^{7} + p^{28} T^{8} \) |
| 61 | $C_2 \wr S_4$ | \( 1 + 2955031 T + 10161069366493 T^{2} + 20511634928342419193 T^{3} + \)\(40\!\cdots\!24\)\( T^{4} + 20511634928342419193 p^{7} T^{5} + 10161069366493 p^{14} T^{6} + 2955031 p^{21} T^{7} + p^{28} T^{8} \) |
| 67 | $C_2 \wr S_4$ | \( 1 - 2737235 T + 11338696552561 T^{2} - 10253324599054769427 T^{3} + \)\(42\!\cdots\!00\)\( T^{4} - 10253324599054769427 p^{7} T^{5} + 11338696552561 p^{14} T^{6} - 2737235 p^{21} T^{7} + p^{28} T^{8} \) |
| 71 | $C_2 \wr S_4$ | \( 1 - 4958184 T + 34530824802083 T^{2} - \)\(13\!\cdots\!88\)\( T^{3} + \)\(46\!\cdots\!44\)\( T^{4} - \)\(13\!\cdots\!88\)\( p^{7} T^{5} + 34530824802083 p^{14} T^{6} - 4958184 p^{21} T^{7} + p^{28} T^{8} \) |
| 73 | $C_2 \wr S_4$ | \( 1 + 931591 T + 36783900926266 T^{2} + 23785151910851054960 T^{3} + \)\(57\!\cdots\!97\)\( T^{4} + 23785151910851054960 p^{7} T^{5} + 36783900926266 p^{14} T^{6} + 931591 p^{21} T^{7} + p^{28} T^{8} \) |
| 79 | $C_2 \wr S_4$ | \( 1 - 5813561 T + 62570152478953 T^{2} - \)\(28\!\cdots\!17\)\( T^{3} + \)\(17\!\cdots\!80\)\( T^{4} - \)\(28\!\cdots\!17\)\( p^{7} T^{5} + 62570152478953 p^{14} T^{6} - 5813561 p^{21} T^{7} + p^{28} T^{8} \) |
| 83 | $C_2 \wr S_4$ | \( 1 - 2120460 T + 82604185739147 T^{2} - \)\(14\!\cdots\!40\)\( T^{3} + \)\(30\!\cdots\!00\)\( T^{4} - \)\(14\!\cdots\!40\)\( p^{7} T^{5} + 82604185739147 p^{14} T^{6} - 2120460 p^{21} T^{7} + p^{28} T^{8} \) |
| 89 | $C_2 \wr S_4$ | \( 1 - 8833716 T + 177304580185760 T^{2} - \)\(11\!\cdots\!56\)\( T^{3} + \)\(11\!\cdots\!98\)\( T^{4} - \)\(11\!\cdots\!56\)\( p^{7} T^{5} + 177304580185760 p^{14} T^{6} - 8833716 p^{21} T^{7} + p^{28} T^{8} \) |
| 97 | $C_2 \wr S_4$ | \( 1 + 22666876 T + 479752420528504 T^{2} + \)\(58\!\cdots\!32\)\( T^{3} + \)\(64\!\cdots\!86\)\( T^{4} + \)\(58\!\cdots\!32\)\( p^{7} T^{5} + 479752420528504 p^{14} T^{6} + 22666876 p^{21} T^{7} + p^{28} T^{8} \) |
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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
−9.962508126897688723390297842405, −9.426949820161535778638724918530, −9.301337930865519238212383203799, −9.221640083167962114200095471210, −9.130898114351751010074685737206, −8.190153499014499833802761633206, −7.975799993867005169480289000987, −7.893191366413452225076333575503, −7.66185920822056925424410226225, −7.31232586276366107171470739243, −6.56161858019835799145226513004, −6.55412350511213497195160477471, −6.46064230065258186579070048093, −5.64506388419974055478535271416, −5.63118640598797215313043541416, −5.55486254918469052860634700054, −5.13599974824715060865115092151, −3.85117549411986779418492347141, −3.58610015018173937621774895127, −3.18869547090167069367072090705, −3.12971676067467901245787159769, −2.09273615838921581829043172705, −1.99323057322219530211809715948, −1.48555689459506694529918385714, −1.09531673000685192307210480991, 0, 0, 0, 0,
1.09531673000685192307210480991, 1.48555689459506694529918385714, 1.99323057322219530211809715948, 2.09273615838921581829043172705, 3.12971676067467901245787159769, 3.18869547090167069367072090705, 3.58610015018173937621774895127, 3.85117549411986779418492347141, 5.13599974824715060865115092151, 5.55486254918469052860634700054, 5.63118640598797215313043541416, 5.64506388419974055478535271416, 6.46064230065258186579070048093, 6.55412350511213497195160477471, 6.56161858019835799145226513004, 7.31232586276366107171470739243, 7.66185920822056925424410226225, 7.893191366413452225076333575503, 7.975799993867005169480289000987, 8.190153499014499833802761633206, 9.130898114351751010074685737206, 9.221640083167962114200095471210, 9.301337930865519238212383203799, 9.426949820161535778638724918530, 9.962508126897688723390297842405
