| L(s) = 1 | − 2·2-s − 5·3-s − 4-s − 6·5-s + 10·6-s − 8·7-s + 5·8-s + 10·9-s + 12·10-s + 5·12-s + 6·13-s + 16·14-s + 30·15-s − 16-s + 4·17-s − 20·18-s + 12·19-s + 6·20-s + 40·21-s + 11·23-s − 25·24-s + 9·25-s − 12·26-s − 6·27-s + 8·28-s + 29-s − 60·30-s + ⋯ |
| L(s) = 1 | − 1.41·2-s − 2.88·3-s − 1/2·4-s − 2.68·5-s + 4.08·6-s − 3.02·7-s + 1.76·8-s + 10/3·9-s + 3.79·10-s + 1.44·12-s + 1.66·13-s + 4.27·14-s + 7.74·15-s − 1/4·16-s + 0.970·17-s − 4.71·18-s + 2.75·19-s + 1.34·20-s + 8.72·21-s + 2.29·23-s − 5.10·24-s + 9/5·25-s − 2.35·26-s − 1.15·27-s + 1.51·28-s + 0.185·29-s − 10.9·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(43^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & -\,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(43^{6}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr=\mathstrut & -\,\Lambda(1-s)\end{aligned}\]
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 | 43 | | \( 1 \) |
| good | 2 | $A_4\times C_2$ | \( 1 + p T + 5 T^{2} + 7 T^{3} + 5 p T^{4} + p^{3} T^{5} + p^{3} T^{6} \) |
| 3 | $A_4\times C_2$ | \( 1 + 5 T + 5 p T^{2} + 31 T^{3} + 5 p^{2} T^{4} + 5 p^{2} T^{5} + p^{3} T^{6} \) |
| 5 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{3} \) |
| 7 | $A_4\times C_2$ | \( 1 + 8 T + 40 T^{2} + 125 T^{3} + 40 p T^{4} + 8 p^{2} T^{5} + p^{3} T^{6} \) |
| 11 | $A_4\times C_2$ | \( 1 + 12 T^{2} - 7 T^{3} + 12 p T^{4} + p^{3} T^{6} \) |
| 13 | $A_4\times C_2$ | \( 1 - 6 T + 44 T^{2} - 157 T^{3} + 44 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 17 | $A_4\times C_2$ | \( 1 - 4 T + 47 T^{2} - 128 T^{3} + 47 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} \) |
| 19 | $A_4\times C_2$ | \( 1 - 12 T + 98 T^{2} - 485 T^{3} + 98 p T^{4} - 12 p^{2} T^{5} + p^{3} T^{6} \) |
| 23 | $A_4\times C_2$ | \( 1 - 11 T + 107 T^{2} - 547 T^{3} + 107 p T^{4} - 11 p^{2} T^{5} + p^{3} T^{6} \) |
| 29 | $A_4\times C_2$ | \( 1 - T + 71 T^{2} - 71 T^{3} + 71 p T^{4} - p^{2} T^{5} + p^{3} T^{6} \) |
| 31 | $A_4\times C_2$ | \( 1 - 5 T + 71 T^{2} - 213 T^{3} + 71 p T^{4} - 5 p^{2} T^{5} + p^{3} T^{6} \) |
| 37 | $A_4\times C_2$ | \( 1 + 2 T + 26 T^{2} + 399 T^{3} + 26 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 41 | $A_4\times C_2$ | \( 1 + 6 T + 86 T^{2} + 311 T^{3} + 86 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 47 | $C_2$ | \( ( 1 + T + p T^{2} )^{3} \) |
| 53 | $A_4\times C_2$ | \( 1 - 9 T + 165 T^{2} - 925 T^{3} + 165 p T^{4} - 9 p^{2} T^{5} + p^{3} T^{6} \) |
| 59 | $A_4\times C_2$ | \( 1 + 24 T + 348 T^{2} + 3169 T^{3} + 348 p T^{4} + 24 p^{2} T^{5} + p^{3} T^{6} \) |
| 61 | $A_4\times C_2$ | \( 1 + T + 139 T^{2} + 205 T^{3} + 139 p T^{4} + p^{2} T^{5} + p^{3} T^{6} \) |
| 67 | $A_4\times C_2$ | \( 1 + 2 T + 172 T^{2} + 281 T^{3} + 172 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 71 | $A_4\times C_2$ | \( 1 + 129 T^{2} - 56 T^{3} + 129 p T^{4} + p^{3} T^{6} \) |
| 73 | $A_4\times C_2$ | \( 1 + 14 T + 212 T^{2} + 1743 T^{3} + 212 p T^{4} + 14 p^{2} T^{5} + p^{3} T^{6} \) |
| 79 | $A_4\times C_2$ | \( 1 - 3 T + 212 T^{2} - 391 T^{3} + 212 p T^{4} - 3 p^{2} T^{5} + p^{3} T^{6} \) |
| 83 | $A_4\times C_2$ | \( 1 - 19 T + 248 T^{2} - 2231 T^{3} + 248 p T^{4} - 19 p^{2} T^{5} + p^{3} T^{6} \) |
| 89 | $A_4\times C_2$ | \( 1 + 26 T + 350 T^{2} + 3439 T^{3} + 350 p T^{4} + 26 p^{2} T^{5} + p^{3} T^{6} \) |
| 97 | $A_4\times C_2$ | \( 1 - 50 T + 1122 T^{2} - 14291 T^{3} + 1122 p T^{4} - 50 p^{2} T^{5} + p^{3} T^{6} \) |
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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
−8.893962551004082014658716696747, −8.281743218602510334975133536582, −7.935161123855898554371246165023, −7.928234464970713260938573064156, −7.56889780451234650073359644510, −7.33586652133484051573307036688, −6.95933191468663916369144050663, −6.68465958982000203984511538073, −6.58345313781237722595343302584, −6.16015407284142689054140029958, −5.87843517579362556606010599211, −5.67771255846854587825780461363, −5.59940103991254503412459215598, −5.01539442397618302576956335891, −4.70442413791124244043115230153, −4.69891771900078336073358832173, −4.16851102936979812298978779660, −3.60176399964199526487480723799, −3.47352585261241276990355651775, −3.30564651331794730935312604812, −3.24190142511258081045431852223, −2.73187757931118196711620680291, −1.19185603502242024754155021776, −1.07399224176184701267095441437, −0.833282867572314989151627316128, 0, 0, 0,
0.833282867572314989151627316128, 1.07399224176184701267095441437, 1.19185603502242024754155021776, 2.73187757931118196711620680291, 3.24190142511258081045431852223, 3.30564651331794730935312604812, 3.47352585261241276990355651775, 3.60176399964199526487480723799, 4.16851102936979812298978779660, 4.69891771900078336073358832173, 4.70442413791124244043115230153, 5.01539442397618302576956335891, 5.59940103991254503412459215598, 5.67771255846854587825780461363, 5.87843517579362556606010599211, 6.16015407284142689054140029958, 6.58345313781237722595343302584, 6.68465958982000203984511538073, 6.95933191468663916369144050663, 7.33586652133484051573307036688, 7.56889780451234650073359644510, 7.928234464970713260938573064156, 7.935161123855898554371246165023, 8.281743218602510334975133536582, 8.893962551004082014658716696747
