| L(s) = 1 | − 2-s − 4-s + 5·5-s − 5·10-s − 2·11-s − 3·13-s + 16-s + 12·17-s + 3·19-s − 5·20-s + 2·22-s + 8·25-s + 3·26-s + 29-s + 3·31-s + 2·32-s − 12·34-s − 3·37-s − 3·38-s + 22·41-s − 3·43-s + 2·44-s + 9·47-s − 8·50-s + 3·52-s − 18·53-s − 10·55-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1/2·4-s + 2.23·5-s − 1.58·10-s − 0.603·11-s − 0.832·13-s + 1/4·16-s + 2.91·17-s + 0.688·19-s − 1.11·20-s + 0.426·22-s + 8/5·25-s + 0.588·26-s + 0.185·29-s + 0.538·31-s + 0.353·32-s − 2.05·34-s − 0.493·37-s − 0.486·38-s + 3.43·41-s − 0.457·43-s + 0.301·44-s + 1.31·47-s − 1.13·50-s + 0.416·52-s − 2.47·53-s − 1.34·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{12} \cdot 7^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{12} \cdot 7^{6}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.810449278\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.810449278\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | | \( 1 \) |
| 7 | | \( 1 \) |
| good | 2 | $S_4\times C_2$ | \( 1 + T + p T^{2} + 3 T^{3} + p^{2} T^{4} + p^{2} T^{5} + p^{3} T^{6} \) |
| 5 | $S_4\times C_2$ | \( 1 - p T + 17 T^{2} - 39 T^{3} + 17 p T^{4} - p^{3} T^{5} + p^{3} T^{6} \) |
| 11 | $S_4\times C_2$ | \( 1 + 2 T + 14 T^{2} - 3 T^{3} + 14 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 13 | $C_2$ | \( ( 1 + T + p T^{2} )^{3} \) |
| 17 | $S_4\times C_2$ | \( 1 - 12 T + 90 T^{2} - 435 T^{3} + 90 p T^{4} - 12 p^{2} T^{5} + p^{3} T^{6} \) |
| 19 | $S_4\times C_2$ | \( 1 - 3 T + 51 T^{2} - 107 T^{3} + 51 p T^{4} - 3 p^{2} T^{5} + p^{3} T^{6} \) |
| 23 | $S_4\times C_2$ | \( 1 + 36 T^{2} - 9 T^{3} + 36 p T^{4} + p^{3} T^{6} \) |
| 29 | $S_4\times C_2$ | \( 1 - T + 83 T^{2} - 57 T^{3} + 83 p T^{4} - p^{2} T^{5} + p^{3} T^{6} \) |
| 31 | $S_4\times C_2$ | \( 1 - 3 T + 69 T^{2} - 213 T^{3} + 69 p T^{4} - 3 p^{2} T^{5} + p^{3} T^{6} \) |
| 37 | $S_4\times C_2$ | \( 1 + 3 T + 57 T^{2} + 303 T^{3} + 57 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} \) |
| 41 | $S_4\times C_2$ | \( 1 - 22 T + 278 T^{2} - 2157 T^{3} + 278 p T^{4} - 22 p^{2} T^{5} + p^{3} T^{6} \) |
| 43 | $S_4\times C_2$ | \( 1 + 3 T + 63 T^{2} + 379 T^{3} + 63 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} \) |
| 47 | $S_4\times C_2$ | \( 1 - 9 T + 87 T^{2} - 657 T^{3} + 87 p T^{4} - 9 p^{2} T^{5} + p^{3} T^{6} \) |
| 53 | $S_4\times C_2$ | \( 1 + 18 T + 234 T^{2} + 1917 T^{3} + 234 p T^{4} + 18 p^{2} T^{5} + p^{3} T^{6} \) |
| 59 | $S_4\times C_2$ | \( 1 - 9 T + 171 T^{2} - 999 T^{3} + 171 p T^{4} - 9 p^{2} T^{5} + p^{3} T^{6} \) |
| 61 | $S_4\times C_2$ | \( 1 - 6 T + 162 T^{2} - 665 T^{3} + 162 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 67 | $S_4\times C_2$ | \( 1 - 6 T^{2} + 683 T^{3} - 6 p T^{4} + p^{3} T^{6} \) |
| 71 | $S_4\times C_2$ | \( 1 - 9 T + 207 T^{2} - 1197 T^{3} + 207 p T^{4} - 9 p^{2} T^{5} + p^{3} T^{6} \) |
| 73 | $S_4\times C_2$ | \( 1 + 3 T + 51 T^{2} + 681 T^{3} + 51 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} \) |
| 79 | $S_4\times C_2$ | \( 1 - 15 T + 189 T^{2} - 1601 T^{3} + 189 p T^{4} - 15 p^{2} T^{5} + p^{3} T^{6} \) |
| 83 | $S_4\times C_2$ | \( 1 - 12 T + 288 T^{2} - 2019 T^{3} + 288 p T^{4} - 12 p^{2} T^{5} + p^{3} T^{6} \) |
| 89 | $S_4\times C_2$ | \( 1 - 2 T + 116 T^{2} - 735 T^{3} + 116 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 97 | $S_4\times C_2$ | \( 1 + 3 T + 177 T^{2} - 21 T^{3} + 177 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} \) |
| show more | | |
| show less | | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{6} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.55375702235774415912876332612, −7.34412457970403397876604625464, −7.19670423749994907545950686853, −6.64214252533158115133029157732, −6.39312857937413224909551460148, −6.04208313840883713592509414739, −5.97721523656946510394674759743, −5.92633241491292663868405926347, −5.41836041405943411281137064347, −5.34061924444122300735878264498, −4.99725113264908268905560638997, −4.84477572420187324365843646499, −4.69968305679906952629261741620, −3.94376487284227679838340412748, −3.84391940630156707376799032610, −3.55014080526982427557507275629, −3.16543783066093183300898347644, −2.78590270549731639406418595633, −2.49772578605917469806824317869, −2.45398168771716383522827811699, −1.88119015681072879994872662474, −1.65556924776515740713041065762, −1.06410307612239049835011446379, −0.879395160376564536799404295835, −0.48722328896480827272175970429,
0.48722328896480827272175970429, 0.879395160376564536799404295835, 1.06410307612239049835011446379, 1.65556924776515740713041065762, 1.88119015681072879994872662474, 2.45398168771716383522827811699, 2.49772578605917469806824317869, 2.78590270549731639406418595633, 3.16543783066093183300898347644, 3.55014080526982427557507275629, 3.84391940630156707376799032610, 3.94376487284227679838340412748, 4.69968305679906952629261741620, 4.84477572420187324365843646499, 4.99725113264908268905560638997, 5.34061924444122300735878264498, 5.41836041405943411281137064347, 5.92633241491292663868405926347, 5.97721523656946510394674759743, 6.04208313840883713592509414739, 6.39312857937413224909551460148, 6.64214252533158115133029157732, 7.19670423749994907545950686853, 7.34412457970403397876604625464, 7.55375702235774415912876332612
