| L(s) = 1 | − 32·2-s + 640·4-s + 100·5-s − 1.37e3·7-s − 1.02e4·8-s − 3.20e3·10-s + 6.44e3·11-s + 6.59e3·13-s + 4.39e4·14-s + 1.43e5·16-s + 2.21e4·17-s − 1.57e4·19-s + 6.40e4·20-s − 2.06e5·22-s − 1.07e5·23-s − 1.38e5·25-s − 2.11e5·26-s − 8.78e5·28-s − 1.63e5·29-s + 3.54e4·31-s − 1.83e6·32-s − 7.09e5·34-s − 1.37e5·35-s + 3.71e5·37-s + 5.03e5·38-s − 1.02e6·40-s + 7.92e4·41-s + ⋯ |
| L(s) = 1 | − 2.82·2-s + 5·4-s + 0.357·5-s − 1.51·7-s − 7.07·8-s − 1.01·10-s + 1.46·11-s + 0.832·13-s + 4.27·14-s + 35/4·16-s + 1.09·17-s − 0.526·19-s + 1.78·20-s − 4.13·22-s − 1.83·23-s − 1.77·25-s − 2.35·26-s − 7.55·28-s − 1.24·29-s + 0.213·31-s − 9.89·32-s − 3.09·34-s − 0.540·35-s + 1.20·37-s + 1.48·38-s − 2.52·40-s + 0.179·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{12} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(8-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{12} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s+7/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(1.321195391\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.321195391\) |
| \(L(\frac{9}{2})\) |
|
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 | $C_1$ | \( ( 1 + p^{3} T )^{4} \) |
| 3 | | \( 1 \) |
| 7 | $C_1$ | \( ( 1 + p^{3} T )^{4} \) |
| good | 5 | $C_2 \wr S_4$ | \( 1 - 4 p^{2} T + 29671 p T^{2} - 732784 p^{2} T^{3} + 3749612 p^{5} T^{4} - 732784 p^{9} T^{5} + 29671 p^{15} T^{6} - 4 p^{23} T^{7} + p^{28} T^{8} \) |
| 11 | $C_2 \wr S_4$ | \( 1 - 586 p T + 37769756 T^{2} - 228514659326 T^{3} + 1078167507581590 T^{4} - 228514659326 p^{7} T^{5} + 37769756 p^{14} T^{6} - 586 p^{22} T^{7} + p^{28} T^{8} \) |
| 13 | $C_2 \wr S_4$ | \( 1 - 6594 T + 19891447 T^{2} - 364106053224 T^{3} + 5942923970112420 T^{4} - 364106053224 p^{7} T^{5} + 19891447 p^{14} T^{6} - 6594 p^{21} T^{7} + p^{28} T^{8} \) |
| 17 | $C_2 \wr S_4$ | \( 1 - 22184 T + 974685890 T^{2} - 17362692213776 T^{3} + 550385287420035907 T^{4} - 17362692213776 p^{7} T^{5} + 974685890 p^{14} T^{6} - 22184 p^{21} T^{7} + p^{28} T^{8} \) |
| 19 | $C_2 \wr S_4$ | \( 1 + 15730 T + 1980440200 T^{2} + 11316965148682 T^{3} + 1806787307706254590 T^{4} + 11316965148682 p^{7} T^{5} + 1980440200 p^{14} T^{6} + 15730 p^{21} T^{7} + p^{28} T^{8} \) |
| 23 | $C_2 \wr S_4$ | \( 1 + 107176 T + 7550967635 T^{2} + 263716086104380 T^{3} + 15096085918285347364 T^{4} + 263716086104380 p^{7} T^{5} + 7550967635 p^{14} T^{6} + 107176 p^{21} T^{7} + p^{28} T^{8} \) |
| 29 | $C_2 \wr S_4$ | \( 1 + 163844 T + 54237375941 T^{2} + 7533654826868990 T^{3} + \)\(13\!\cdots\!96\)\( T^{4} + 7533654826868990 p^{7} T^{5} + 54237375941 p^{14} T^{6} + 163844 p^{21} T^{7} + p^{28} T^{8} \) |
| 31 | $C_2 \wr S_4$ | \( 1 - 35424 T + 78920739349 T^{2} - 2293037033326794 T^{3} + \)\(28\!\cdots\!76\)\( T^{4} - 2293037033326794 p^{7} T^{5} + 78920739349 p^{14} T^{6} - 35424 p^{21} T^{7} + p^{28} T^{8} \) |
| 37 | $C_2 \wr S_4$ | \( 1 - 371678 T + 36654646105 T^{2} + 4638931743405058 T^{3} - 27536034361369275620 T^{4} + 4638931743405058 p^{7} T^{5} + 36654646105 p^{14} T^{6} - 371678 p^{21} T^{7} + p^{28} T^{8} \) |
| 41 | $C_2 \wr S_4$ | \( 1 - 1932 p T + 601991836139 T^{2} - 27938884514442804 T^{3} + \)\(16\!\cdots\!80\)\( T^{4} - 27938884514442804 p^{7} T^{5} + 601991836139 p^{14} T^{6} - 1932 p^{22} T^{7} + p^{28} T^{8} \) |
| 43 | $C_2 \wr S_4$ | \( 1 - 1186652 T + 1017715522390 T^{2} - 641363373474832928 T^{3} + \)\(35\!\cdots\!11\)\( T^{4} - 641363373474832928 p^{7} T^{5} + 1017715522390 p^{14} T^{6} - 1186652 p^{21} T^{7} + p^{28} T^{8} \) |
| 47 | $C_2 \wr S_4$ | \( 1 + 952110 T + 1004903355581 T^{2} + 895872494580890022 T^{3} + \)\(78\!\cdots\!80\)\( T^{4} + 895872494580890022 p^{7} T^{5} + 1004903355581 p^{14} T^{6} + 952110 p^{21} T^{7} + p^{28} T^{8} \) |
| 53 | $C_2 \wr S_4$ | \( 1 + 155082 T + 3887296735253 T^{2} + 501972673899779730 T^{3} + \)\(64\!\cdots\!88\)\( T^{4} + 501972673899779730 p^{7} T^{5} + 3887296735253 p^{14} T^{6} + 155082 p^{21} T^{7} + p^{28} T^{8} \) |
| 59 | $C_2 \wr S_4$ | \( 1 - 1193848 T + 9190200912434 T^{2} - 7545670251918781024 T^{3} + \)\(32\!\cdots\!07\)\( T^{4} - 7545670251918781024 p^{7} T^{5} + 9190200912434 p^{14} T^{6} - 1193848 p^{21} T^{7} + p^{28} T^{8} \) |
| 61 | $C_2 \wr S_4$ | \( 1 - 129102 T + 3936176501464 T^{2} + 69943666393612350 p T^{3} + \)\(89\!\cdots\!86\)\( T^{4} + 69943666393612350 p^{8} T^{5} + 3936176501464 p^{14} T^{6} - 129102 p^{21} T^{7} + p^{28} T^{8} \) |
| 67 | $C_2 \wr S_4$ | \( 1 + 2555300 T + 10181443348363 T^{2} + 19491175790322912344 T^{3} + \)\(44\!\cdots\!52\)\( T^{4} + 19491175790322912344 p^{7} T^{5} + 10181443348363 p^{14} T^{6} + 2555300 p^{21} T^{7} + p^{28} T^{8} \) |
| 71 | $C_2 \wr S_4$ | \( 1 + 2975872 T + 37134034488479 T^{2} + 79994336615079452740 T^{3} + \)\(50\!\cdots\!76\)\( T^{4} + 79994336615079452740 p^{7} T^{5} + 37134034488479 p^{14} T^{6} + 2975872 p^{21} T^{7} + p^{28} T^{8} \) |
| 73 | $C_2 \wr S_4$ | \( 1 - 6428020 T + 50916284945092 T^{2} - \)\(18\!\cdots\!60\)\( T^{3} + \)\(83\!\cdots\!30\)\( T^{4} - \)\(18\!\cdots\!60\)\( p^{7} T^{5} + 50916284945092 p^{14} T^{6} - 6428020 p^{21} T^{7} + p^{28} T^{8} \) |
| 79 | $C_2 \wr S_4$ | \( 1 - 6843240 T + 65675461015531 T^{2} - \)\(31\!\cdots\!80\)\( T^{3} + \)\(18\!\cdots\!96\)\( T^{4} - \)\(31\!\cdots\!80\)\( p^{7} T^{5} + 65675461015531 p^{14} T^{6} - 6843240 p^{21} T^{7} + p^{28} T^{8} \) |
| 83 | $C_2 \wr S_4$ | \( 1 + 11642074 T + 59294184135017 T^{2} + 2971274369590018450 T^{3} - \)\(81\!\cdots\!40\)\( T^{4} + 2971274369590018450 p^{7} T^{5} + 59294184135017 p^{14} T^{6} + 11642074 p^{21} T^{7} + p^{28} T^{8} \) |
| 89 | $C_2 \wr S_4$ | \( 1 + 21148548 T + 217796908793279 T^{2} + \)\(15\!\cdots\!32\)\( T^{3} + \)\(98\!\cdots\!40\)\( T^{4} + \)\(15\!\cdots\!32\)\( p^{7} T^{5} + 217796908793279 p^{14} T^{6} + 21148548 p^{21} T^{7} + p^{28} T^{8} \) |
| 97 | $C_2 \wr S_4$ | \( 1 + 5112362 T + 272857671132172 T^{2} + \)\(11\!\cdots\!66\)\( T^{3} + \)\(31\!\cdots\!98\)\( T^{4} + \)\(11\!\cdots\!66\)\( p^{7} T^{5} + 272857671132172 p^{14} T^{6} + 5112362 p^{21} T^{7} + p^{28} T^{8} \) |
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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
−7.30984780438033769713079782837, −6.57794648827270022381987597635, −6.54063128872071111038504834710, −6.50744131135849368777025742610, −6.33297638504614776852674248305, −5.75418868527228306995364723246, −5.69114166004511492686058800842, −5.56347439440697506983328472777, −5.48338489603775862511323943584, −4.31390409415444999667876340125, −4.25759526667851314169524694693, −4.06195065687028061595659086419, −3.87268131391226795567985449949, −3.36382215895760881750141401225, −3.10961444712394152516210479793, −2.85557603734613413381867516026, −2.66075923588026349360422184892, −1.95613913880126226820121245275, −1.92976105554027109338120472581, −1.66537941057250679613885884466, −1.55236892342212453675406932816, −0.866131940808010814934886001261, −0.74053174513455142519499868410, −0.44755902954282307304486467502, −0.30611204246770947099127205904,
0.30611204246770947099127205904, 0.44755902954282307304486467502, 0.74053174513455142519499868410, 0.866131940808010814934886001261, 1.55236892342212453675406932816, 1.66537941057250679613885884466, 1.92976105554027109338120472581, 1.95613913880126226820121245275, 2.66075923588026349360422184892, 2.85557603734613413381867516026, 3.10961444712394152516210479793, 3.36382215895760881750141401225, 3.87268131391226795567985449949, 4.06195065687028061595659086419, 4.25759526667851314169524694693, 4.31390409415444999667876340125, 5.48338489603775862511323943584, 5.56347439440697506983328472777, 5.69114166004511492686058800842, 5.75418868527228306995364723246, 6.33297638504614776852674248305, 6.50744131135849368777025742610, 6.54063128872071111038504834710, 6.57794648827270022381987597635, 7.30984780438033769713079782837
