| L(s) = 1 | − 89·3-s − 5.25e3·7-s + 3.55e3·9-s + 5.46e4·11-s + 2.15e5·13-s + 3.34e5·17-s − 8.18e5·19-s + 4.67e5·21-s − 3.52e6·23-s − 1.70e6·27-s + 2.17e6·29-s − 4.27e6·31-s − 4.86e6·33-s − 1.03e7·37-s − 1.92e7·39-s + 5.92e6·41-s + 2.44e7·43-s − 6.68e7·47-s − 4.99e7·49-s − 2.98e7·51-s + 1.32e8·53-s + 7.28e7·57-s − 5.67e6·59-s + 1.25e8·61-s − 1.86e7·63-s − 8.88e7·67-s + 3.13e8·69-s + ⋯ |
| L(s) = 1 | − 0.634·3-s − 0.827·7-s + 0.180·9-s + 1.12·11-s + 2.09·13-s + 0.972·17-s − 1.44·19-s + 0.525·21-s − 2.62·23-s − 0.618·27-s + 0.571·29-s − 0.831·31-s − 0.714·33-s − 0.903·37-s − 1.32·39-s + 0.327·41-s + 1.08·43-s − 1.99·47-s − 1.23·49-s − 0.617·51-s + 2.30·53-s + 0.914·57-s − 0.0609·59-s + 1.15·61-s − 0.149·63-s − 0.538·67-s + 1.66·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 64000000 ^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr =\mathstrut & -\, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 64000000 ^{s/2} \, \Gamma_{\C}(s+9/2)^{3} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | | \( 1 \) |
| 5 | | \( 1 \) |
| good | 3 | $S_4\times C_2$ | \( 1 + 89 T + 1456 p T^{2} + 197885 p^{2} T^{3} + 1456 p^{10} T^{4} + 89 p^{18} T^{5} + p^{27} T^{6} \) |
| 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
−9.068780995190921582844332468672, −8.507193459332937899851008712978, −8.362713020275378917848261601968, −8.346781262949214263032713438068, −7.73779896662537267021876872511, −7.56782408312553048388910504912, −6.84721939836532501575337329560, −6.66032113257674828790778057928, −6.57442666274241545224730869107, −6.17831575250707882051588714239, −5.80433075120218912925796937517, −5.68496534398194478323240842752, −5.46045456486930213646119293111, −4.75973405008141794157062305000, −4.38758986166924061250061789829, −3.97820064675609500420387973137, −3.81220822044965051013463761184, −3.58318522449777500164746017189, −3.31606791697050035242525933616, −2.64323624852184597378757473320, −2.19221982310895030789353597593, −1.94260534857480988070618899603, −1.37453309741623086856223524531, −1.20238427993438059216118919276, −0.923339878482621828947555191737, 0, 0, 0,
0.923339878482621828947555191737, 1.20238427993438059216118919276, 1.37453309741623086856223524531, 1.94260534857480988070618899603, 2.19221982310895030789353597593, 2.64323624852184597378757473320, 3.31606791697050035242525933616, 3.58318522449777500164746017189, 3.81220822044965051013463761184, 3.97820064675609500420387973137, 4.38758986166924061250061789829, 4.75973405008141794157062305000, 5.46045456486930213646119293111, 5.68496534398194478323240842752, 5.80433075120218912925796937517, 6.17831575250707882051588714239, 6.57442666274241545224730869107, 6.66032113257674828790778057928, 6.84721939836532501575337329560, 7.56782408312553048388910504912, 7.73779896662537267021876872511, 8.346781262949214263032713438068, 8.362713020275378917848261601968, 8.507193459332937899851008712978, 9.068780995190921582844332468672
