| L(s) = 1 | − 33·2-s − 53·4-s + 5.25e3·7-s + 1.93e4·8-s + 5.46e4·11-s + 2.15e5·13-s − 1.73e5·14-s − 5.02e5·16-s − 3.34e5·17-s + 8.18e5·19-s − 1.80e6·22-s − 3.52e6·23-s − 7.12e6·26-s − 2.78e5·28-s − 2.17e6·29-s + 4.27e6·31-s + 6.15e6·32-s + 1.10e7·34-s − 1.03e7·37-s − 2.70e7·38-s − 5.92e6·41-s − 2.44e7·43-s − 2.89e6·44-s + 1.16e8·46-s − 6.68e7·47-s − 4.99e7·49-s − 1.14e7·52-s + ⋯ |
| L(s) = 1 | − 1.45·2-s − 0.103·4-s + 0.827·7-s + 1.67·8-s + 1.12·11-s + 2.09·13-s − 1.20·14-s − 1.91·16-s − 0.972·17-s + 1.44·19-s − 1.64·22-s − 2.62·23-s − 3.05·26-s − 0.0856·28-s − 0.571·29-s + 0.831·31-s + 1.03·32-s + 1.41·34-s − 0.903·37-s − 2.10·38-s − 0.327·41-s − 1.08·43-s − 0.116·44-s + 3.83·46-s − 1.99·47-s − 1.23·49-s − 0.217·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 11390625 ^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr =\mathstrut & -\, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 11390625 ^{s/2} \, \Gamma_{\C}(s+9/2)^{3} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{11}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | | \( 1 \) |
| 5 | | \( 1 \) |
| good | 2 | $S_4\times C_2$ | \( 1 + 33 T + 571 p T^{2} + 2505 p^{3} T^{3} + 571 p^{10} T^{4} + 33 p^{18} T^{5} + p^{27} T^{6} \) |
| 7 | $S_4\times C_2$ | \( 1 - 5258 T + 11082951 p T^{2} - 9176976100 p^{2} T^{3} + 11082951 p^{10} T^{4} - 5258 p^{18} T^{5} + p^{27} T^{6} \) |
| 11 | $S_4\times C_2$ | \( 1 - 54699 T + 5816187440 T^{2} - 182671409832855 T^{3} + 5816187440 p^{9} T^{4} - 54699 p^{18} T^{5} + p^{27} T^{6} \) |
| 13 | $S_4\times C_2$ | \( 1 - 215884 T + 31225156343 T^{2} - 3094983572287480 T^{3} + 31225156343 p^{9} T^{4} - 215884 p^{18} T^{5} + p^{27} T^{6} \) |
| 17 | $S_4\times C_2$ | \( 1 + 334983 T + 355030662062 T^{2} + 77675997792847395 T^{3} + 355030662062 p^{9} T^{4} + 334983 p^{18} T^{5} + p^{27} T^{6} \) |
| 19 | $S_4\times C_2$ | \( 1 - 818845 T + 468173656712 T^{2} - 302339390932836385 T^{3} + 468173656712 p^{9} T^{4} - 818845 p^{18} T^{5} + p^{27} T^{6} \) |
| 23 | $S_4\times C_2$ | \( 1 + 3526854 T + 8700596724993 T^{2} + 13086552252942250620 T^{3} + 8700596724993 p^{9} T^{4} + 3526854 p^{18} T^{5} + p^{27} T^{6} \) |
| 29 | $S_4\times C_2$ | \( 1 + 2175480 T + 1119231270883 p T^{2} + 59155485309271560240 T^{3} + 1119231270883 p^{10} T^{4} + 2175480 p^{18} T^{5} + p^{27} T^{6} \) |
| 31 | $S_4\times C_2$ | \( 1 - 4274066 T + 82851493809465 T^{2} - \)\(22\!\cdots\!20\)\( T^{3} + 82851493809465 p^{9} T^{4} - 4274066 p^{18} T^{5} + p^{27} T^{6} \) |
| 37 | $S_4\times C_2$ | \( 1 + 10305042 T + 340842196290747 T^{2} + \)\(25\!\cdots\!40\)\( T^{3} + 340842196290747 p^{9} T^{4} + 10305042 p^{18} T^{5} + p^{27} T^{6} \) |
| 41 | $S_4\times C_2$ | \( 1 + 5926311 T + 232043727124790 T^{2} + \)\(23\!\cdots\!95\)\( T^{3} + 232043727124790 p^{9} T^{4} + 5926311 p^{18} T^{5} + p^{27} T^{6} \) |
| 43 | $S_4\times C_2$ | \( 1 + 24429956 T + 1676050268074193 T^{2} + \)\(57\!\cdots\!00\)\( p T^{3} + 1676050268074193 p^{9} T^{4} + 24429956 p^{18} T^{5} + p^{27} T^{6} \) |
| 47 | $S_4\times C_2$ | \( 1 + 66858708 T + 4517266130065277 T^{2} + \)\(15\!\cdots\!80\)\( T^{3} + 4517266130065277 p^{9} T^{4} + 66858708 p^{18} T^{5} + p^{27} T^{6} \) |
| 53 | $S_4\times C_2$ | \( 1 + 132620514 T + 9209248487675243 T^{2} + \)\(47\!\cdots\!40\)\( T^{3} + 9209248487675243 p^{9} T^{4} + 132620514 p^{18} T^{5} + p^{27} T^{6} \) |
| 59 | $S_4\times C_2$ | \( 1 + 5670960 T + 19337695182532817 T^{2} + \)\(14\!\cdots\!80\)\( T^{3} + 19337695182532817 p^{9} T^{4} + 5670960 p^{18} T^{5} + p^{27} T^{6} \) |
| 61 | $S_4\times C_2$ | \( 1 - 125306926 T + 22192130203358915 T^{2} - \)\(23\!\cdots\!20\)\( T^{3} + 22192130203358915 p^{9} T^{4} - 125306926 p^{18} T^{5} + p^{27} T^{6} \) |
| 67 | $S_4\times C_2$ | \( 1 - 88829483 T + 30562384919894712 T^{2} - \)\(76\!\cdots\!95\)\( T^{3} + 30562384919894712 p^{9} T^{4} - 88829483 p^{18} T^{5} + p^{27} T^{6} \) |
| 71 | $S_4\times C_2$ | \( 1 + 297550596 T + 87036096018332165 T^{2} + \)\(15\!\cdots\!20\)\( T^{3} + 87036096018332165 p^{9} T^{4} + 297550596 p^{18} T^{5} + p^{27} T^{6} \) |
| 73 | $S_4\times C_2$ | \( 1 - 181321729 T + 94365252526618118 T^{2} - \)\(75\!\cdots\!45\)\( T^{3} + 94365252526618118 p^{9} T^{4} - 181321729 p^{18} T^{5} + p^{27} T^{6} \) |
| 79 | $S_4\times C_2$ | \( 1 + 310025170 T + 176092553119892457 T^{2} + \)\(48\!\cdots\!60\)\( T^{3} + 176092553119892457 p^{9} T^{4} + 310025170 p^{18} T^{5} + p^{27} T^{6} \) |
| 83 | $S_4\times C_2$ | \( 1 - 731088801 T + 685433348047337568 T^{2} - \)\(27\!\cdots\!65\)\( T^{3} + 685433348047337568 p^{9} T^{4} - 731088801 p^{18} T^{5} + p^{27} T^{6} \) |
| 89 | $S_4\times C_2$ | \( 1 - 1103860035 T + 1320664213408319502 T^{2} - \)\(75\!\cdots\!55\)\( T^{3} + 1320664213408319502 p^{9} T^{4} - 1103860035 p^{18} T^{5} + p^{27} T^{6} \) |
| 97 | $S_4\times C_2$ | \( 1 + 332236842 T + 1487745026246658927 T^{2} + \)\(16\!\cdots\!20\)\( T^{3} + 1487745026246658927 p^{9} T^{4} + 332236842 p^{18} T^{5} + p^{27} T^{6} \) |
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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
−9.729127705039457534062396830252, −9.378021008256606264115327914315, −9.196949995703184222544275330090, −8.858583588639534514465544341421, −8.281633024535546041212606427660, −8.278250579033802037400483610843, −8.265565027417019727554645599484, −7.66488257817239271409657595139, −7.38766332202424494197309483302, −6.69051931267212043080931701107, −6.49378759640728048416024136575, −6.12883236225042766208396542237, −6.03908214428889447679923157067, −5.02705266084094745873021869789, −4.87507819520758355109507379577, −4.86358504029189244303985546156, −3.98552106258928298433732962218, −3.78428718913426423094006692701, −3.64846138124535386496785135987, −3.07985710292661232301959906668, −2.11747004833917081373575597136, −2.06509623634391592728097012695, −1.35814053088191203594240637721, −1.25474505947154774559700749520, −1.09783233397848597953313942335, 0, 0, 0,
1.09783233397848597953313942335, 1.25474505947154774559700749520, 1.35814053088191203594240637721, 2.06509623634391592728097012695, 2.11747004833917081373575597136, 3.07985710292661232301959906668, 3.64846138124535386496785135987, 3.78428718913426423094006692701, 3.98552106258928298433732962218, 4.86358504029189244303985546156, 4.87507819520758355109507379577, 5.02705266084094745873021869789, 6.03908214428889447679923157067, 6.12883236225042766208396542237, 6.49378759640728048416024136575, 6.69051931267212043080931701107, 7.38766332202424494197309483302, 7.66488257817239271409657595139, 8.265565027417019727554645599484, 8.278250579033802037400483610843, 8.281633024535546041212606427660, 8.858583588639534514465544341421, 9.196949995703184222544275330090, 9.378021008256606264115327914315, 9.729127705039457534062396830252
