| L(s) = 1 | − 3·2-s + 3·3-s + 6·4-s − 9·6-s − 2·7-s − 10·8-s + 6·9-s + 18·12-s − 4·13-s + 6·14-s + 15·16-s − 6·17-s − 18·18-s − 4·19-s − 6·21-s − 3·23-s − 30·24-s + 12·26-s + 10·27-s − 12·28-s + 14·29-s − 12·31-s − 21·32-s + 18·34-s + 36·36-s − 22·37-s + 12·38-s + ⋯ |
| L(s) = 1 | − 2.12·2-s + 1.73·3-s + 3·4-s − 3.67·6-s − 0.755·7-s − 3.53·8-s + 2·9-s + 5.19·12-s − 1.10·13-s + 1.60·14-s + 15/4·16-s − 1.45·17-s − 4.24·18-s − 0.917·19-s − 1.30·21-s − 0.625·23-s − 6.12·24-s + 2.35·26-s + 1.92·27-s − 2.26·28-s + 2.59·29-s − 2.15·31-s − 3.71·32-s + 3.08·34-s + 6·36-s − 3.61·37-s + 1.94·38-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{3} \cdot 3^{3} \cdot 5^{6} \cdot 23^{3}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & -\,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{3} \cdot 3^{3} \cdot 5^{6} \cdot 23^{3}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr=\mathstrut & -\,\Lambda(1-s)\end{aligned}\]
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 | $C_1$ | \( ( 1 + T )^{3} \) |
| 3 | $C_1$ | \( ( 1 - T )^{3} \) |
| 5 | | \( 1 \) |
| 23 | $C_1$ | \( ( 1 + T )^{3} \) |
| good | 7 | $S_4\times C_2$ | \( 1 + 2 T + 9 T^{2} + 20 T^{3} + 9 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 11 | $S_4\times C_2$ | \( 1 + 17 T^{2} - 16 T^{3} + 17 p T^{4} + p^{3} T^{6} \) |
| 13 | $S_4\times C_2$ | \( 1 + 4 T + 31 T^{2} + 88 T^{3} + 31 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} \) |
| 17 | $S_4\times C_2$ | \( 1 + 6 T + 47 T^{2} + 196 T^{3} + 47 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 19 | $S_4\times C_2$ | \( 1 + 4 T + 49 T^{2} + 136 T^{3} + 49 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} \) |
| 29 | $D_{6}$ | \( 1 - 14 T + 91 T^{2} - 468 T^{3} + 91 p T^{4} - 14 p^{2} T^{5} + p^{3} T^{6} \) |
| 31 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{3} \) |
| 37 | $S_4\times C_2$ | \( 1 + 22 T + 251 T^{2} + 1860 T^{3} + 251 p T^{4} + 22 p^{2} T^{5} + p^{3} T^{6} \) |
| 41 | $S_4\times C_2$ | \( 1 + 6 T + 71 T^{2} + 500 T^{3} + 71 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 43 | $S_4\times C_2$ | \( 1 + 10 T + 125 T^{2} + 852 T^{3} + 125 p T^{4} + 10 p^{2} T^{5} + p^{3} T^{6} \) |
| 47 | $S_4\times C_2$ | \( 1 - 8 T + 77 T^{2} - 496 T^{3} + 77 p T^{4} - 8 p^{2} T^{5} + p^{3} T^{6} \) |
| 53 | $S_4\times C_2$ | \( 1 + 10 T + 179 T^{2} + 1052 T^{3} + 179 p T^{4} + 10 p^{2} T^{5} + p^{3} T^{6} \) |
| 59 | $S_4\times C_2$ | \( 1 - 2 T + 93 T^{2} - 132 T^{3} + 93 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 61 | $S_4\times C_2$ | \( 1 + 14 T + 115 T^{2} + 596 T^{3} + 115 p T^{4} + 14 p^{2} T^{5} + p^{3} T^{6} \) |
| 67 | $S_4\times C_2$ | \( 1 + 22 T + 341 T^{2} + 3180 T^{3} + 341 p T^{4} + 22 p^{2} T^{5} + p^{3} T^{6} \) |
| 71 | $S_4\times C_2$ | \( 1 - 6 T + 89 T^{2} - 1084 T^{3} + 89 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 73 | $S_4\times C_2$ | \( 1 + 8 T + 187 T^{2} + 1040 T^{3} + 187 p T^{4} + 8 p^{2} T^{5} + p^{3} T^{6} \) |
| 79 | $S_4\times C_2$ | \( 1 + 4 T + 93 T^{2} + 40 T^{3} + 93 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} \) |
| 83 | $S_4\times C_2$ | \( 1 + 20 T + 361 T^{2} + 3480 T^{3} + 361 p T^{4} + 20 p^{2} T^{5} + p^{3} T^{6} \) |
| 89 | $S_4\times C_2$ | \( 1 - 8 T + 235 T^{2} - 1296 T^{3} + 235 p T^{4} - 8 p^{2} T^{5} + p^{3} T^{6} \) |
| 97 | $S_4\times C_2$ | \( 1 - 6 T + 183 T^{2} - 916 T^{3} + 183 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \) |
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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
−8.283013254583847062558932925473, −7.62009069527465232948193517951, −7.60506886814523222289369793541, −7.43676034073385431128546156090, −7.13190996567842549811751982294, −6.87638009373953689059645306842, −6.68180735587254452461593073657, −6.36346294452675799415739035381, −6.32297653828020260530502818724, −6.06324685912821118957369484650, −5.32076779685019756064998667478, −5.18873584588930821924570432981, −4.99962534891339095194888744620, −4.54692251220821180241167425042, −4.11315054453020328279282118304, −4.08303523085363037730409887917, −3.47077496089904345466788332906, −3.19016260270778973526500073578, −3.17972308835723140094762879940, −2.69632030699141062047286179774, −2.34525011462089560304973847153, −2.28196805559722431665587707675, −1.61446291174717628369372889735, −1.58437240387044442982921760080, −1.36373858304239881915024214646, 0, 0, 0,
1.36373858304239881915024214646, 1.58437240387044442982921760080, 1.61446291174717628369372889735, 2.28196805559722431665587707675, 2.34525011462089560304973847153, 2.69632030699141062047286179774, 3.17972308835723140094762879940, 3.19016260270778973526500073578, 3.47077496089904345466788332906, 4.08303523085363037730409887917, 4.11315054453020328279282118304, 4.54692251220821180241167425042, 4.99962534891339095194888744620, 5.18873584588930821924570432981, 5.32076779685019756064998667478, 6.06324685912821118957369484650, 6.32297653828020260530502818724, 6.36346294452675799415739035381, 6.68180735587254452461593073657, 6.87638009373953689059645306842, 7.13190996567842549811751982294, 7.43676034073385431128546156090, 7.60506886814523222289369793541, 7.62009069527465232948193517951, 8.283013254583847062558932925473
