| L(s) = 1 | + 3·5-s − 4·7-s − 8·11-s − 2·13-s − 2·17-s + 8·19-s + 6·25-s + 3·29-s + 4·31-s − 12·35-s − 10·37-s − 10·41-s + 14·43-s + 10·47-s − 49-s − 10·53-s − 24·55-s − 8·59-s − 22·61-s − 6·65-s − 12·67-s + 8·71-s − 2·73-s + 32·77-s − 12·83-s − 6·85-s − 18·89-s + ⋯ |
| L(s) = 1 | + 1.34·5-s − 1.51·7-s − 2.41·11-s − 0.554·13-s − 0.485·17-s + 1.83·19-s + 6/5·25-s + 0.557·29-s + 0.718·31-s − 2.02·35-s − 1.64·37-s − 1.56·41-s + 2.13·43-s + 1.45·47-s − 1/7·49-s − 1.37·53-s − 3.23·55-s − 1.04·59-s − 2.81·61-s − 0.744·65-s − 1.46·67-s + 0.949·71-s − 0.234·73-s + 3.64·77-s − 1.31·83-s − 0.650·85-s − 1.90·89-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{6} \cdot 3^{6} \cdot 5^{3} \cdot 29^{3}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & -\,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{6} \cdot 3^{6} \cdot 5^{3} \cdot 29^{3}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr=\mathstrut & -\,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 3 | | \( 1 \) |
| 5 | $C_1$ | \( ( 1 - T )^{3} \) |
| 29 | $C_1$ | \( ( 1 - T )^{3} \) |
| good | 7 | $S_4\times C_2$ | \( 1 + 4 T + 17 T^{2} + 52 T^{3} + 17 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} \) |
| 11 | $S_4\times C_2$ | \( 1 + 8 T + 45 T^{2} + 164 T^{3} + 45 p T^{4} + 8 p^{2} T^{5} + p^{3} T^{6} \) |
| 13 | $S_4\times C_2$ | \( 1 + 2 T + 19 T^{2} + 28 T^{3} + 19 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 17 | $S_4\times C_2$ | \( 1 + 2 T + 15 T^{2} - 40 T^{3} + 15 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 19 | $S_4\times C_2$ | \( 1 - 8 T + 41 T^{2} - 140 T^{3} + 41 p T^{4} - 8 p^{2} T^{5} + p^{3} T^{6} \) |
| 23 | $S_4\times C_2$ | \( 1 + 45 T^{2} + 36 T^{3} + 45 p T^{4} + p^{3} T^{6} \) |
| 31 | $S_4\times C_2$ | \( 1 - 4 T + 61 T^{2} - 284 T^{3} + 61 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} \) |
| 37 | $S_4\times C_2$ | \( 1 + 10 T + 59 T^{2} + 232 T^{3} + 59 p T^{4} + 10 p^{2} T^{5} + p^{3} T^{6} \) |
| 41 | $S_4\times C_2$ | \( 1 + 10 T + 135 T^{2} + 796 T^{3} + 135 p T^{4} + 10 p^{2} T^{5} + p^{3} T^{6} \) |
| 43 | $D_{6}$ | \( 1 - 14 T + 113 T^{2} - 656 T^{3} + 113 p T^{4} - 14 p^{2} T^{5} + p^{3} T^{6} \) |
| 47 | $S_4\times C_2$ | \( 1 - 10 T + 165 T^{2} - 952 T^{3} + 165 p T^{4} - 10 p^{2} T^{5} + p^{3} T^{6} \) |
| 53 | $S_4\times C_2$ | \( 1 + 10 T + 147 T^{2} + 988 T^{3} + 147 p T^{4} + 10 p^{2} T^{5} + p^{3} T^{6} \) |
| 59 | $S_4\times C_2$ | \( 1 + 8 T + 9 T^{2} - 544 T^{3} + 9 p T^{4} + 8 p^{2} T^{5} + p^{3} T^{6} \) |
| 61 | $D_{6}$ | \( 1 + 22 T + 299 T^{2} + 2788 T^{3} + 299 p T^{4} + 22 p^{2} T^{5} + p^{3} T^{6} \) |
| 67 | $S_4\times C_2$ | \( 1 + 12 T + 225 T^{2} + 1540 T^{3} + 225 p T^{4} + 12 p^{2} T^{5} + p^{3} T^{6} \) |
| 71 | $S_4\times C_2$ | \( 1 - 8 T + 189 T^{2} - 992 T^{3} + 189 p T^{4} - 8 p^{2} T^{5} + p^{3} T^{6} \) |
| 73 | $S_4\times C_2$ | \( 1 + 2 T + 139 T^{2} - 32 T^{3} + 139 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 79 | $S_4\times C_2$ | \( 1 + 129 T^{2} + 244 T^{3} + 129 p T^{4} + p^{3} T^{6} \) |
| 83 | $S_4\times C_2$ | \( 1 + 12 T + 213 T^{2} + 1884 T^{3} + 213 p T^{4} + 12 p^{2} T^{5} + p^{3} T^{6} \) |
| 89 | $S_4\times C_2$ | \( 1 + 18 T + 279 T^{2} + 3132 T^{3} + 279 p T^{4} + 18 p^{2} T^{5} + p^{3} T^{6} \) |
| 97 | $S_4\times C_2$ | \( 1 + 38 T + 763 T^{2} + 9280 T^{3} + 763 p T^{4} + 38 p^{2} T^{5} + p^{3} T^{6} \) |
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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
−7.59856827358761333804860977260, −7.35118778127331763399440308823, −7.14200907682236930242847764227, −6.92687192791166052311481522380, −6.56033229582637809809321863363, −6.34785680134332272186749732621, −6.26439082449963576541415048603, −5.95375970594210402773554882291, −5.51592119291581419662411843173, −5.42716492047027328077659318231, −5.26283226594296558626517902941, −5.01405023583652197521142122963, −4.92846684637236090465011592352, −4.34484038441904270340928245055, −4.07560427281879546221822867948, −3.99316201191640441343857693944, −3.31387031175211363420609494670, −3.05966648970538958709406427185, −2.95193618032618234049668460411, −2.68416228837162531807896907641, −2.54456662855585889951573625539, −2.34865965233774095571105588383, −1.44087116678613878655112741439, −1.42096228406856806185233043419, −1.27692092150067006038620236935, 0, 0, 0,
1.27692092150067006038620236935, 1.42096228406856806185233043419, 1.44087116678613878655112741439, 2.34865965233774095571105588383, 2.54456662855585889951573625539, 2.68416228837162531807896907641, 2.95193618032618234049668460411, 3.05966648970538958709406427185, 3.31387031175211363420609494670, 3.99316201191640441343857693944, 4.07560427281879546221822867948, 4.34484038441904270340928245055, 4.92846684637236090465011592352, 5.01405023583652197521142122963, 5.26283226594296558626517902941, 5.42716492047027328077659318231, 5.51592119291581419662411843173, 5.95375970594210402773554882291, 6.26439082449963576541415048603, 6.34785680134332272186749732621, 6.56033229582637809809321863363, 6.92687192791166052311481522380, 7.14200907682236930242847764227, 7.35118778127331763399440308823, 7.59856827358761333804860977260
