| L(s) = 1 | + 48·2-s + 156·3-s + 1.53e3·4-s − 1.27e3·5-s + 7.48e3·6-s + 1.70e4·7-s + 4.09e4·8-s + 3.78e3·9-s − 6.10e4·10-s + 7.39e4·11-s + 2.39e5·12-s − 8.56e4·13-s + 8.18e5·14-s − 1.98e5·15-s + 9.83e5·16-s + 3.74e5·17-s + 1.81e5·18-s + 4.18e5·19-s − 1.95e6·20-s + 2.66e6·21-s + 3.55e6·22-s + 1.02e6·23-s + 6.38e6·24-s − 4.52e5·25-s − 4.11e6·26-s − 2.93e5·27-s + 2.62e7·28-s + ⋯ |
| L(s) = 1 | + 2.12·2-s + 1.11·3-s + 3·4-s − 0.910·5-s + 2.35·6-s + 2.68·7-s + 3.53·8-s + 0.192·9-s − 1.93·10-s + 1.52·11-s + 3.33·12-s − 0.832·13-s + 5.69·14-s − 1.01·15-s + 15/4·16-s + 1.08·17-s + 0.407·18-s + 0.736·19-s − 2.73·20-s + 2.98·21-s + 3.23·22-s + 0.764·23-s + 3.93·24-s − 0.231·25-s − 1.76·26-s − 0.106·27-s + 8.05·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 17576 ^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr =\mathstrut & \, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 17576 ^{s/2} \, \Gamma_{\C}(s+9/2)^{3} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(\approx\) |
\(30.87980167\) |
| \(L(\frac12)\) |
\(\approx\) |
\(30.87980167\) |
| \(L(\frac{11}{2})\) |
|
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 | $C_1$ | \( ( 1 - p^{4} T )^{3} \) |
| 13 | $C_1$ | \( ( 1 + p^{4} T )^{3} \) |
| good | 3 | $S_4\times C_2$ | \( 1 - 52 p T + 2284 p^{2} T^{2} - 86072 p^{3} T^{3} + 2284 p^{11} T^{4} - 52 p^{19} T^{5} + p^{27} T^{6} \) |
| 5 | $S_4\times C_2$ | \( 1 + 1272 T + 2070036 T^{2} + 393966854 p T^{3} + 2070036 p^{9} T^{4} + 1272 p^{18} T^{5} + p^{27} T^{6} \) |
| 7 | $S_4\times C_2$ | \( 1 - 17058 T + 26151072 p T^{2} - 188573075434 p T^{3} + 26151072 p^{10} T^{4} - 17058 p^{18} T^{5} + p^{27} T^{6} \) |
| 11 | $S_4\times C_2$ | \( 1 - 73974 T + 2778732765 T^{2} - 44497153312780 T^{3} + 2778732765 p^{9} T^{4} - 73974 p^{18} T^{5} + p^{27} T^{6} \) |
| 17 | $S_4\times C_2$ | \( 1 - 374976 T + 378452421528 T^{2} - 86535045026636022 T^{3} + 378452421528 p^{9} T^{4} - 374976 p^{18} T^{5} + p^{27} T^{6} \) |
| 19 | $S_4\times C_2$ | \( 1 - 418338 T + 678538095285 T^{2} - 294307734083078340 T^{3} + 678538095285 p^{9} T^{4} - 418338 p^{18} T^{5} + p^{27} T^{6} \) |
| 23 | $S_4\times C_2$ | \( 1 - 44616 p T + 4031709543477 T^{2} - 2277298243304136464 T^{3} + 4031709543477 p^{9} T^{4} - 44616 p^{19} T^{5} + p^{27} T^{6} \) |
| 29 | $S_4\times C_2$ | \( 1 + 3075834 T + 33419160920259 T^{2} + 58301045097658242844 T^{3} + 33419160920259 p^{9} T^{4} + 3075834 p^{18} T^{5} + p^{27} T^{6} \) |
| 31 | $S_4\times C_2$ | \( 1 - 9286482 T + 43584325295901 T^{2} - \)\(12\!\cdots\!48\)\( T^{3} + 43584325295901 p^{9} T^{4} - 9286482 p^{18} T^{5} + p^{27} T^{6} \) |
| 37 | $S_4\times C_2$ | \( 1 + 17647776 T + 422551891884468 T^{2} + \)\(12\!\cdots\!86\)\( p T^{3} + 422551891884468 p^{9} T^{4} + 17647776 p^{18} T^{5} + p^{27} T^{6} \) |
| 41 | $S_4\times C_2$ | \( 1 + 47257110 T + 1622226515041563 T^{2} + \)\(32\!\cdots\!20\)\( T^{3} + 1622226515041563 p^{9} T^{4} + 47257110 p^{18} T^{5} + p^{27} T^{6} \) |
| 43 | $S_4\times C_2$ | \( 1 + 60023760 T + 2518841308328124 T^{2} + \)\(63\!\cdots\!60\)\( T^{3} + 2518841308328124 p^{9} T^{4} + 60023760 p^{18} T^{5} + p^{27} T^{6} \) |
| 47 | $S_4\times C_2$ | \( 1 + 40824726 T + 3473474836314168 T^{2} + \)\(85\!\cdots\!22\)\( T^{3} + 3473474836314168 p^{9} T^{4} + 40824726 p^{18} T^{5} + p^{27} T^{6} \) |
| 53 | $S_4\times C_2$ | \( 1 - 8072046 T + 660804482335551 T^{2} + \)\(30\!\cdots\!36\)\( T^{3} + 660804482335551 p^{9} T^{4} - 8072046 p^{18} T^{5} + p^{27} T^{6} \) |
| 59 | $S_4\times C_2$ | \( 1 - 138035310 T + 23544326414292237 T^{2} - \)\(17\!\cdots\!80\)\( T^{3} + 23544326414292237 p^{9} T^{4} - 138035310 p^{18} T^{5} + p^{27} T^{6} \) |
| 61 | $S_4\times C_2$ | \( 1 - 264203886 T + 55185039025738935 T^{2} - \)\(65\!\cdots\!40\)\( T^{3} + 55185039025738935 p^{9} T^{4} - 264203886 p^{18} T^{5} + p^{27} T^{6} \) |
| 67 | $S_4\times C_2$ | \( 1 - 203167074 T + 11045547237099813 T^{2} - \)\(71\!\cdots\!08\)\( T^{3} + 11045547237099813 p^{9} T^{4} - 203167074 p^{18} T^{5} + p^{27} T^{6} \) |
| 71 | $S_4\times C_2$ | \( 1 - 123067110 T + 95038593276035448 T^{2} - \)\(13\!\cdots\!70\)\( T^{3} + 95038593276035448 p^{9} T^{4} - 123067110 p^{18} T^{5} + p^{27} T^{6} \) |
| 73 | $S_4\times C_2$ | \( 1 + 433013250 T + 184996030152367959 T^{2} + \)\(44\!\cdots\!00\)\( T^{3} + 184996030152367959 p^{9} T^{4} + 433013250 p^{18} T^{5} + p^{27} T^{6} \) |
| 79 | $S_4\times C_2$ | \( 1 - 406418748 T - 6408821024487075 T^{2} + \)\(51\!\cdots\!80\)\( T^{3} - 6408821024487075 p^{9} T^{4} - 406418748 p^{18} T^{5} + p^{27} T^{6} \) |
| 83 | $S_4\times C_2$ | \( 1 - 105365610 T + 274290745881973209 T^{2} + \)\(14\!\cdots\!40\)\( T^{3} + 274290745881973209 p^{9} T^{4} - 105365610 p^{18} T^{5} + p^{27} T^{6} \) |
| 89 | $S_4\times C_2$ | \( 1 - 1365375798 T + 1535126512833912615 T^{2} - \)\(10\!\cdots\!80\)\( T^{3} + 1535126512833912615 p^{9} T^{4} - 1365375798 p^{18} T^{5} + p^{27} T^{6} \) |
| 97 | $S_4\times C_2$ | \( 1 - 669691662 T + 2079640297453933599 T^{2} - \)\(10\!\cdots\!72\)\( T^{3} + 2079640297453933599 p^{9} T^{4} - 669691662 p^{18} T^{5} + p^{27} T^{6} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{6} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−14.09608740655576413427351920610, −13.30916462258319402621133614747, −13.17042006463907150093165583249, −12.18542563286740613441891397132, −11.74426485930354517952432725776, −11.72121643594694407665657597751, −11.63317550986234936906998739574, −11.02457257974585874400892714978, −10.01137448474895768068270841404, −10.00351134446347953848748770680, −8.765180152378539314120936209563, −8.226121991309560324682467827931, −8.187495588100124948500267043616, −7.58834681285322131875642754176, −6.83862118865833794464806274186, −6.71549113291660741542005804989, −5.37758605401187071670608207917, −5.03915247665482653560310856420, −4.85246768591779478963194471394, −4.05200569128137294835610265933, −3.34582053097380242518917170494, −3.31099108046416242459972015361, −1.98865148009664826600856808029, −1.77943665152290458193534898090, −1.01716492383021255831880708701,
1.01716492383021255831880708701, 1.77943665152290458193534898090, 1.98865148009664826600856808029, 3.31099108046416242459972015361, 3.34582053097380242518917170494, 4.05200569128137294835610265933, 4.85246768591779478963194471394, 5.03915247665482653560310856420, 5.37758605401187071670608207917, 6.71549113291660741542005804989, 6.83862118865833794464806274186, 7.58834681285322131875642754176, 8.187495588100124948500267043616, 8.226121991309560324682467827931, 8.765180152378539314120936209563, 10.00351134446347953848748770680, 10.01137448474895768068270841404, 11.02457257974585874400892714978, 11.63317550986234936906998739574, 11.72121643594694407665657597751, 11.74426485930354517952432725776, 12.18542563286740613441891397132, 13.17042006463907150093165583249, 13.30916462258319402621133614747, 14.09608740655576413427351920610
