| L(s) = 1 | + 6·2-s − 108·3-s − 35·4-s + 276·5-s − 648·6-s + 1.11e3·7-s − 588·8-s + 7.29e3·9-s + 1.65e3·10-s − 4.96e3·11-s + 3.78e3·12-s + 6.69e3·14-s − 2.98e4·15-s + 7.54e3·16-s + 2.29e4·17-s + 4.37e4·18-s + 2.66e4·19-s − 9.66e3·20-s − 1.20e5·21-s − 2.98e4·22-s + 2.37e4·23-s + 6.35e4·24-s − 1.21e5·25-s − 3.93e5·27-s − 3.90e4·28-s + 2.27e5·29-s − 1.78e5·30-s + ⋯ |
| L(s) = 1 | + 0.530·2-s − 2.30·3-s − 0.273·4-s + 0.987·5-s − 1.22·6-s + 1.22·7-s − 0.406·8-s + 10/3·9-s + 0.523·10-s − 1.12·11-s + 0.631·12-s + 0.652·14-s − 2.28·15-s + 0.460·16-s + 1.13·17-s + 1.76·18-s + 0.890·19-s − 0.270·20-s − 2.84·21-s − 0.596·22-s + 0.406·23-s + 0.937·24-s − 1.55·25-s − 3.84·27-s − 0.336·28-s + 1.73·29-s − 1.20·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{4} \cdot 13^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(8-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{4} \cdot 13^{8}\right)^{s/2} \, \Gamma_{\C}(s+7/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(0.07726600893\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.07726600893\) |
| \(L(\frac{9}{2})\) |
|
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 | $C_1$ | \( ( 1 + p^{3} T )^{4} \) |
| 13 | | \( 1 \) |
| good | 2 | $C_2 \wr S_4$ | \( 1 - 3 p T + 71 T^{2} - 3 p^{4} T^{3} - 2075 p^{2} T^{4} - 3 p^{11} T^{5} + 71 p^{14} T^{6} - 3 p^{22} T^{7} + p^{28} T^{8} \) |
| 5 | $C_2 \wr S_4$ | \( 1 - 276 T + 197788 T^{2} - 40498476 T^{3} + 3835865934 p T^{4} - 40498476 p^{7} T^{5} + 197788 p^{14} T^{6} - 276 p^{21} T^{7} + p^{28} T^{8} \) |
| 7 | $C_2 \wr S_4$ | \( 1 - 1116 T + 1289364 T^{2} - 1475837100 T^{3} + 1725189275510 T^{4} - 1475837100 p^{7} T^{5} + 1289364 p^{14} T^{6} - 1116 p^{21} T^{7} + p^{28} T^{8} \) |
| 11 | $C_2 \wr S_4$ | \( 1 + 4968 T + 31789764 T^{2} + 7700715864 p T^{3} + 2321489535302 p^{2} T^{4} + 7700715864 p^{8} T^{5} + 31789764 p^{14} T^{6} + 4968 p^{21} T^{7} + p^{28} T^{8} \) |
| 17 | $C_2 \wr S_4$ | \( 1 - 22944 T + 46359436 p T^{2} - 8949696696480 T^{3} + 320251989994944934 T^{4} - 8949696696480 p^{7} T^{5} + 46359436 p^{15} T^{6} - 22944 p^{21} T^{7} + p^{28} T^{8} \) |
| 19 | $C_2 \wr S_4$ | \( 1 - 26628 T + 1840429956 T^{2} - 1864274027748 T^{3} + 55956940394342402 p T^{4} - 1864274027748 p^{7} T^{5} + 1840429956 p^{14} T^{6} - 26628 p^{21} T^{7} + p^{28} T^{8} \) |
| 23 | $C_2 \wr S_4$ | \( 1 - 23712 T + 5676492540 T^{2} + 252664112684256 T^{3} + 7341963438813530342 T^{4} + 252664112684256 p^{7} T^{5} + 5676492540 p^{14} T^{6} - 23712 p^{21} T^{7} + p^{28} T^{8} \) |
| 29 | $C_2 \wr S_4$ | \( 1 - 7848 p T + 62614978092 T^{2} - 9668001029233176 T^{3} + \)\(16\!\cdots\!18\)\( T^{4} - 9668001029233176 p^{7} T^{5} + 62614978092 p^{14} T^{6} - 7848 p^{22} T^{7} + p^{28} T^{8} \) |
| 31 | $C_2 \wr S_4$ | \( 1 + 531380 T + 160244933124 T^{2} + 33147973279904644 T^{3} + \)\(58\!\cdots\!82\)\( T^{4} + 33147973279904644 p^{7} T^{5} + 160244933124 p^{14} T^{6} + 531380 p^{21} T^{7} + p^{28} T^{8} \) |
| 37 | $C_2 \wr S_4$ | \( 1 - 630032 T + 363024898524 T^{2} - 115966414486757488 T^{3} + \)\(42\!\cdots\!90\)\( T^{4} - 115966414486757488 p^{7} T^{5} + 363024898524 p^{14} T^{6} - 630032 p^{21} T^{7} + p^{28} T^{8} \) |
| 41 | $C_2 \wr S_4$ | \( 1 + 212028 T + 610756221260 T^{2} + 104248024230193908 T^{3} + \)\(16\!\cdots\!26\)\( T^{4} + 104248024230193908 p^{7} T^{5} + 610756221260 p^{14} T^{6} + 212028 p^{21} T^{7} + p^{28} T^{8} \) |
| 43 | $C_2 \wr S_4$ | \( 1 + 1883816 T + 2074933849452 T^{2} + 1593838758492466792 T^{3} + \)\(94\!\cdots\!62\)\( T^{4} + 1593838758492466792 p^{7} T^{5} + 2074933849452 p^{14} T^{6} + 1883816 p^{21} T^{7} + p^{28} T^{8} \) |
| 47 | $C_2 \wr S_4$ | \( 1 - 218976 T + 1674287804260 T^{2} - 275086024439887392 T^{3} + \)\(12\!\cdots\!78\)\( T^{4} - 275086024439887392 p^{7} T^{5} + 1674287804260 p^{14} T^{6} - 218976 p^{21} T^{7} + p^{28} T^{8} \) |
| 53 | $C_2 \wr S_4$ | \( 1 - 4469064 T + 11600239729644 T^{2} - 20062469835544503576 T^{3} + \)\(25\!\cdots\!14\)\( T^{4} - 20062469835544503576 p^{7} T^{5} + 11600239729644 p^{14} T^{6} - 4469064 p^{21} T^{7} + p^{28} T^{8} \) |
| 59 | $C_2 \wr S_4$ | \( 1 + 6869472 T + 26833463094068 T^{2} + 68413421569825980576 T^{3} + \)\(12\!\cdots\!62\)\( T^{4} + 68413421569825980576 p^{7} T^{5} + 26833463094068 p^{14} T^{6} + 6869472 p^{21} T^{7} + p^{28} T^{8} \) |
| 61 | $C_2 \wr S_4$ | \( 1 - 458456 T - 3618122605428 T^{2} + 106326846041522936 T^{3} + \)\(20\!\cdots\!70\)\( T^{4} + 106326846041522936 p^{7} T^{5} - 3618122605428 p^{14} T^{6} - 458456 p^{21} T^{7} + p^{28} T^{8} \) |
| 67 | $C_2 \wr S_4$ | \( 1 - 2218700 T + 15862796529780 T^{2} - 24248895987437470252 T^{3} + \)\(12\!\cdots\!46\)\( T^{4} - 24248895987437470252 p^{7} T^{5} + 15862796529780 p^{14} T^{6} - 2218700 p^{21} T^{7} + p^{28} T^{8} \) |
| 71 | $C_2 \wr S_4$ | \( 1 + 2473128 T + 16425758339700 T^{2} + 41038598687138174952 T^{3} + \)\(24\!\cdots\!82\)\( T^{4} + 41038598687138174952 p^{7} T^{5} + 16425758339700 p^{14} T^{6} + 2473128 p^{21} T^{7} + p^{28} T^{8} \) |
| 73 | $C_2 \wr S_4$ | \( 1 + 10457616 T + 76660569461196 T^{2} + \)\(38\!\cdots\!16\)\( T^{3} + \)\(14\!\cdots\!50\)\( T^{4} + \)\(38\!\cdots\!16\)\( p^{7} T^{5} + 76660569461196 p^{14} T^{6} + 10457616 p^{21} T^{7} + p^{28} T^{8} \) |
| 79 | $C_2 \wr S_4$ | \( 1 - 5130864 T + 35855585806908 T^{2} - \)\(11\!\cdots\!76\)\( T^{3} + \)\(66\!\cdots\!90\)\( T^{4} - \)\(11\!\cdots\!76\)\( p^{7} T^{5} + 35855585806908 p^{14} T^{6} - 5130864 p^{21} T^{7} + p^{28} T^{8} \) |
| 83 | $C_2 \wr S_4$ | \( 1 + 16493520 T + 168902969196404 T^{2} + \)\(11\!\cdots\!64\)\( T^{3} + \)\(66\!\cdots\!10\)\( T^{4} + \)\(11\!\cdots\!64\)\( p^{7} T^{5} + 168902969196404 p^{14} T^{6} + 16493520 p^{21} T^{7} + p^{28} T^{8} \) |
| 89 | $C_2 \wr S_4$ | \( 1 - 3209484 T + 133604223233500 T^{2} - \)\(38\!\cdots\!80\)\( T^{3} + \)\(82\!\cdots\!38\)\( T^{4} - \)\(38\!\cdots\!80\)\( p^{7} T^{5} + 133604223233500 p^{14} T^{6} - 3209484 p^{21} T^{7} + p^{28} T^{8} \) |
| 97 | $C_2 \wr S_4$ | \( 1 + 17620768 T + 324352574382924 T^{2} + \)\(37\!\cdots\!40\)\( T^{3} + \)\(39\!\cdots\!14\)\( T^{4} + \)\(37\!\cdots\!40\)\( p^{7} T^{5} + 324352574382924 p^{14} T^{6} + 17620768 p^{21} T^{7} + p^{28} T^{8} \) |
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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
−6.66449530558738861629135472366, −6.24961503393935999456369259022, −6.00234524051272269003381934022, −5.78088708120749935948701456711, −5.69591692808014965604607225352, −5.51684729961483270757415198148, −5.22155269770303983080526222629, −5.08684250543701808571350052981, −5.06657841852814327926745179402, −4.38166543205333381630922677790, −4.36537456856712328856952236433, −4.29716447651317678962945380212, −3.78232202825059121225658174067, −3.42625778938511740491503039780, −3.14942111483131408256857376184, −2.89742845397958117143137692158, −2.57884402056205973078256039994, −2.11583458062948322946666677680, −1.74429869190530098160836726386, −1.57435269960269125760606384857, −1.40102863799338031243820260518, −1.00223990508567955590362328109, −0.971651396938054115834802386405, −0.24964750041301705896520416108, −0.05425691281086791241870405989,
0.05425691281086791241870405989, 0.24964750041301705896520416108, 0.971651396938054115834802386405, 1.00223990508567955590362328109, 1.40102863799338031243820260518, 1.57435269960269125760606384857, 1.74429869190530098160836726386, 2.11583458062948322946666677680, 2.57884402056205973078256039994, 2.89742845397958117143137692158, 3.14942111483131408256857376184, 3.42625778938511740491503039780, 3.78232202825059121225658174067, 4.29716447651317678962945380212, 4.36537456856712328856952236433, 4.38166543205333381630922677790, 5.06657841852814327926745179402, 5.08684250543701808571350052981, 5.22155269770303983080526222629, 5.51684729961483270757415198148, 5.69591692808014965604607225352, 5.78088708120749935948701456711, 6.00234524051272269003381934022, 6.24961503393935999456369259022, 6.66449530558738861629135472366
