| L(s) = 1 | − 4·2-s − 195·4-s + 500·5-s − 1.37e3·7-s + 968·8-s − 2.00e3·10-s − 3.03e3·11-s + 952·13-s + 5.48e3·14-s + 309·16-s + 5.04e4·17-s − 1.56e4·19-s − 9.75e4·20-s + 1.21e4·22-s + 8.96e4·23-s + 1.56e5·25-s − 3.80e3·26-s + 2.67e5·28-s + 1.52e5·29-s − 1.74e5·31-s − 3.18e3·32-s − 2.01e5·34-s − 6.86e5·35-s + 3.36e5·37-s + 6.26e4·38-s + 4.84e5·40-s + 8.31e5·41-s + ⋯ |
| L(s) = 1 | − 0.353·2-s − 1.52·4-s + 1.78·5-s − 1.51·7-s + 0.668·8-s − 0.632·10-s − 0.686·11-s + 0.120·13-s + 0.534·14-s + 0.0188·16-s + 2.49·17-s − 0.524·19-s − 2.72·20-s + 0.242·22-s + 1.53·23-s + 2·25-s − 0.0424·26-s + 2.30·28-s + 1.15·29-s − 1.05·31-s − 0.0171·32-s − 0.880·34-s − 2.70·35-s + 1.09·37-s + 0.185·38-s + 1.19·40-s + 1.88·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 5^{4} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(8-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 5^{4} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s+7/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(5.194139575\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.194139575\) |
| \(L(\frac{9}{2})\) |
|
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 | | \( 1 \) |
| 5 | $C_1$ | \( ( 1 - p^{3} T )^{4} \) |
| 7 | $C_1$ | \( ( 1 + p^{3} T )^{4} \) |
| good | 2 | $C_2 \wr S_4$ | \( 1 + p^{2} T + 211 T^{2} + 41 p^{4} T^{3} + 9897 p^{2} T^{4} + 41 p^{11} T^{5} + 211 p^{14} T^{6} + p^{23} T^{7} + p^{28} T^{8} \) |
| 11 | $C_2 \wr S_4$ | \( 1 + 3032 T + 8724556 T^{2} - 9607116968 T^{3} - 435979430040906 T^{4} - 9607116968 p^{7} T^{5} + 8724556 p^{14} T^{6} + 3032 p^{21} T^{7} + p^{28} T^{8} \) |
| 13 | $C_2 \wr S_4$ | \( 1 - 952 T + 210831564 T^{2} - 26422048232 T^{3} + 18428705438718038 T^{4} - 26422048232 p^{7} T^{5} + 210831564 p^{14} T^{6} - 952 p^{21} T^{7} + p^{28} T^{8} \) |
| 17 | $C_2 \wr S_4$ | \( 1 - 50464 T + 1960031020 T^{2} - 2707317464992 p T^{3} + 1066565176033263846 T^{4} - 2707317464992 p^{8} T^{5} + 1960031020 p^{14} T^{6} - 50464 p^{21} T^{7} + p^{28} T^{8} \) |
| 19 | $C_2 \wr S_4$ | \( 1 + 15672 T + 559905228 T^{2} + 2014359873960 p T^{3} + 941509224802953686 T^{4} + 2014359873960 p^{8} T^{5} + 559905228 p^{14} T^{6} + 15672 p^{21} T^{7} + p^{28} T^{8} \) |
| 23 | $C_2 \wr S_4$ | \( 1 - 89656 T + 4501325212 T^{2} + 69687633399784 T^{3} - 14152756109912601690 T^{4} + 69687633399784 p^{7} T^{5} + 4501325212 p^{14} T^{6} - 89656 p^{21} T^{7} + p^{28} T^{8} \) |
| 29 | $C_2 \wr S_4$ | \( 1 - 152032 T + 21473699900 T^{2} + 2815235970024608 T^{3} - \)\(38\!\cdots\!46\)\( T^{4} + 2815235970024608 p^{7} T^{5} + 21473699900 p^{14} T^{6} - 152032 p^{21} T^{7} + p^{28} T^{8} \) |
| 31 | $C_2 \wr S_4$ | \( 1 + 174768 T + 55891141180 T^{2} + 3637198052560944 T^{3} + \)\(13\!\cdots\!98\)\( T^{4} + 3637198052560944 p^{7} T^{5} + 55891141180 p^{14} T^{6} + 174768 p^{21} T^{7} + p^{28} T^{8} \) |
| 37 | $C_2 \wr S_4$ | \( 1 - 336624 T + 92883209980 T^{2} - 39975242594106384 T^{3} + \)\(23\!\cdots\!06\)\( T^{4} - 39975242594106384 p^{7} T^{5} + 92883209980 p^{14} T^{6} - 336624 p^{21} T^{7} + p^{28} T^{8} \) |
| 41 | $C_2 \wr S_4$ | \( 1 - 831320 T + 573127348636 T^{2} - 181506743184224680 T^{3} + \)\(87\!\cdots\!46\)\( T^{4} - 181506743184224680 p^{7} T^{5} + 573127348636 p^{14} T^{6} - 831320 p^{21} T^{7} + p^{28} T^{8} \) |
| 43 | $C_2 \wr S_4$ | \( 1 - 487304 T + 471477191052 T^{2} - 161247674046157384 T^{3} + \)\(17\!\cdots\!70\)\( T^{4} - 161247674046157384 p^{7} T^{5} + 471477191052 p^{14} T^{6} - 487304 p^{21} T^{7} + p^{28} T^{8} \) |
| 47 | $C_2 \wr S_4$ | \( 1 - 1694344 T + 1687980559676 T^{2} - 1587664985783234536 T^{3} + \)\(13\!\cdots\!58\)\( T^{4} - 1587664985783234536 p^{7} T^{5} + 1687980559676 p^{14} T^{6} - 1694344 p^{21} T^{7} + p^{28} T^{8} \) |
| 53 | $C_2 \wr S_4$ | \( 1 - 1622704 T + 4508893590556 T^{2} - 4582392561753692432 T^{3} + \)\(75\!\cdots\!70\)\( T^{4} - 4582392561753692432 p^{7} T^{5} + 4508893590556 p^{14} T^{6} - 1622704 p^{21} T^{7} + p^{28} T^{8} \) |
| 59 | $C_2 \wr S_4$ | \( 1 - 1791904 T + 6532959030380 T^{2} - 153373839518454752 p T^{3} + \)\(19\!\cdots\!78\)\( T^{4} - 153373839518454752 p^{8} T^{5} + 6532959030380 p^{14} T^{6} - 1791904 p^{21} T^{7} + p^{28} T^{8} \) |
| 61 | $C_2 \wr S_4$ | \( 1 + 6305568 T + 22780335900988 T^{2} + 55492549833610211616 T^{3} + \)\(10\!\cdots\!90\)\( T^{4} + 55492549833610211616 p^{7} T^{5} + 22780335900988 p^{14} T^{6} + 6305568 p^{21} T^{7} + p^{28} T^{8} \) |
| 67 | $C_2 \wr S_4$ | \( 1 + 5721352 T + 23595305900076 T^{2} + 75571221219427221896 T^{3} + \)\(21\!\cdots\!54\)\( T^{4} + 75571221219427221896 p^{7} T^{5} + 23595305900076 p^{14} T^{6} + 5721352 p^{21} T^{7} + p^{28} T^{8} \) |
| 71 | $C_2 \wr S_4$ | \( 1 - 806800 T + 20155152959804 T^{2} + 15073864590324028400 T^{3} + \)\(18\!\cdots\!66\)\( T^{4} + 15073864590324028400 p^{7} T^{5} + 20155152959804 p^{14} T^{6} - 806800 p^{21} T^{7} + p^{28} T^{8} \) |
| 73 | $C_2 \wr S_4$ | \( 1 + 2063992 T + 35421614540700 T^{2} + 62409042187233168968 T^{3} + \)\(54\!\cdots\!66\)\( T^{4} + 62409042187233168968 p^{7} T^{5} + 35421614540700 p^{14} T^{6} + 2063992 p^{21} T^{7} + p^{28} T^{8} \) |
| 79 | $C_2 \wr S_4$ | \( 1 + 9897216 T + 96954696383292 T^{2} + \)\(51\!\cdots\!68\)\( T^{3} + \)\(28\!\cdots\!10\)\( T^{4} + \)\(51\!\cdots\!68\)\( p^{7} T^{5} + 96954696383292 p^{14} T^{6} + 9897216 p^{21} T^{7} + p^{28} T^{8} \) |
| 83 | $C_2 \wr S_4$ | \( 1 + 5170384 T + 70407618174284 T^{2} + \)\(27\!\cdots\!08\)\( T^{3} + \)\(26\!\cdots\!14\)\( T^{4} + \)\(27\!\cdots\!08\)\( p^{7} T^{5} + 70407618174284 p^{14} T^{6} + 5170384 p^{21} T^{7} + p^{28} T^{8} \) |
| 89 | $C_2 \wr S_4$ | \( 1 - 12345528 T + 180750343764508 T^{2} - \)\(14\!\cdots\!40\)\( T^{3} + \)\(12\!\cdots\!66\)\( T^{4} - \)\(14\!\cdots\!40\)\( p^{7} T^{5} + 180750343764508 p^{14} T^{6} - 12345528 p^{21} T^{7} + p^{28} T^{8} \) |
| 97 | $C_2 \wr S_4$ | \( 1 - 13936104 T + 198193159922940 T^{2} - \)\(20\!\cdots\!44\)\( T^{3} + \)\(23\!\cdots\!66\)\( T^{4} - \)\(20\!\cdots\!44\)\( p^{7} T^{5} + 198193159922940 p^{14} T^{6} - 13936104 p^{21} T^{7} + p^{28} T^{8} \) |
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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
−7.31007014230634680479851869985, −6.88777741653067880410313849145, −6.82871151696415169390679356341, −6.27422463394007922274497398052, −6.07606182156998626289142226947, −5.83420352007061922317428107096, −5.60856264868042493587990654961, −5.47932624234387837268007658377, −5.47005618877209568551117436933, −4.62233869744241909775488058371, −4.46280937420096757264392200480, −4.40378916127145478529069286544, −4.34828524032710868204742891102, −3.42279480739007068341152528755, −3.41223640283924579526147926490, −3.03969365025535124716053328708, −2.90073084993387940017573125007, −2.45662367712936081203689038223, −2.25346662672633875927096551511, −1.85493690299370276416057422499, −1.38606627780954706214194499993, −0.891921022649445418425196044003, −0.869089727176070460892081697141, −0.59099261967129068709472973859, −0.32995984817426760118821690634,
0.32995984817426760118821690634, 0.59099261967129068709472973859, 0.869089727176070460892081697141, 0.891921022649445418425196044003, 1.38606627780954706214194499993, 1.85493690299370276416057422499, 2.25346662672633875927096551511, 2.45662367712936081203689038223, 2.90073084993387940017573125007, 3.03969365025535124716053328708, 3.41223640283924579526147926490, 3.42279480739007068341152528755, 4.34828524032710868204742891102, 4.40378916127145478529069286544, 4.46280937420096757264392200480, 4.62233869744241909775488058371, 5.47005618877209568551117436933, 5.47932624234387837268007658377, 5.60856264868042493587990654961, 5.83420352007061922317428107096, 6.07606182156998626289142226947, 6.27422463394007922274497398052, 6.82871151696415169390679356341, 6.88777741653067880410313849145, 7.31007014230634680479851869985
