| L(s) = 1 | − 3·3-s − 4·5-s + 10·7-s + 6·9-s − 11-s + 12·15-s − 7·17-s + 11·19-s − 30·21-s − 2·23-s − 2·25-s − 10·27-s − 8·29-s + 8·31-s + 3·33-s − 40·35-s − 14·37-s − 41-s + 3·43-s − 24·45-s − 9·47-s + 48·49-s + 21·51-s − 13·53-s + 4·55-s − 33·57-s − 14·59-s + ⋯ |
| L(s) = 1 | − 1.73·3-s − 1.78·5-s + 3.77·7-s + 2·9-s − 0.301·11-s + 3.09·15-s − 1.69·17-s + 2.52·19-s − 6.54·21-s − 0.417·23-s − 2/5·25-s − 1.92·27-s − 1.48·29-s + 1.43·31-s + 0.522·33-s − 6.76·35-s − 2.30·37-s − 0.156·41-s + 0.457·43-s − 3.57·45-s − 1.31·47-s + 48/7·49-s + 2.94·51-s − 1.78·53-s + 0.539·55-s − 4.37·57-s − 1.82·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 3^{3} \cdot 13^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & -\,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 3^{3} \cdot 13^{6}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr=\mathstrut & -\,\Lambda(1-s)\end{aligned}\]
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 | 2 | | \( 1 \) |
| 3 | $C_1$ | \( ( 1 + T )^{3} \) |
| 13 | | \( 1 \) |
| good | 5 | $A_4\times C_2$ | \( 1 + 4 T + 18 T^{2} + 39 T^{3} + 18 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} \) |
| 7 | $A_4\times C_2$ | \( 1 - 10 T + 52 T^{2} - 169 T^{3} + 52 p T^{4} - 10 p^{2} T^{5} + p^{3} T^{6} \) |
| 11 | $A_4\times C_2$ | \( 1 + T + 3 T^{2} - 21 T^{3} + 3 p T^{4} + p^{2} T^{5} + p^{3} T^{6} \) |
| 17 | $A_4\times C_2$ | \( 1 + 7 T + 65 T^{2} + 245 T^{3} + 65 p T^{4} + 7 p^{2} T^{5} + p^{3} T^{6} \) |
| 19 | $A_4\times C_2$ | \( 1 - 11 T + 67 T^{2} - 305 T^{3} + 67 p T^{4} - 11 p^{2} T^{5} + p^{3} T^{6} \) |
| 23 | $A_4\times C_2$ | \( 1 + 2 T + 26 T^{2} + 175 T^{3} + 26 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 29 | $A_4\times C_2$ | \( 1 + 8 T + 92 T^{2} + 421 T^{3} + 92 p T^{4} + 8 p^{2} T^{5} + p^{3} T^{6} \) |
| 31 | $A_4\times C_2$ | \( 1 - 8 T + 70 T^{2} - 299 T^{3} + 70 p T^{4} - 8 p^{2} T^{5} + p^{3} T^{6} \) |
| 37 | $A_4\times C_2$ | \( 1 + 14 T + 174 T^{2} + 1127 T^{3} + 174 p T^{4} + 14 p^{2} T^{5} + p^{3} T^{6} \) |
| 41 | $A_4\times C_2$ | \( 1 + T + 121 T^{2} + 81 T^{3} + 121 p T^{4} + p^{2} T^{5} + p^{3} T^{6} \) |
| 43 | $A_4\times C_2$ | \( 1 - 3 T + 104 T^{2} - 287 T^{3} + 104 p T^{4} - 3 p^{2} T^{5} + p^{3} T^{6} \) |
| 47 | $A_4\times C_2$ | \( 1 + 9 T + 21 T^{2} - 65 T^{3} + 21 p T^{4} + 9 p^{2} T^{5} + p^{3} T^{6} \) |
| 53 | $A_4\times C_2$ | \( 1 + 13 T + 199 T^{2} + 1407 T^{3} + 199 p T^{4} + 13 p^{2} T^{5} + p^{3} T^{6} \) |
| 59 | $A_4\times C_2$ | \( 1 + 14 T + 3 p T^{2} + 1596 T^{3} + 3 p^{2} T^{4} + 14 p^{2} T^{5} + p^{3} T^{6} \) |
| 61 | $A_4\times C_2$ | \( 1 + 13 T + 195 T^{2} + 1363 T^{3} + 195 p T^{4} + 13 p^{2} T^{5} + p^{3} T^{6} \) |
| 67 | $A_4\times C_2$ | \( 1 - 5 T + 179 T^{2} - 573 T^{3} + 179 p T^{4} - 5 p^{2} T^{5} + p^{3} T^{6} \) |
| 71 | $A_4\times C_2$ | \( 1 + 6 T + 134 T^{2} + 391 T^{3} + 134 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 73 | $A_4\times C_2$ | \( 1 + 18 T + 320 T^{2} + 2795 T^{3} + 320 p T^{4} + 18 p^{2} T^{5} + p^{3} T^{6} \) |
| 79 | $A_4\times C_2$ | \( 1 - 9 T + 215 T^{2} - 1253 T^{3} + 215 p T^{4} - 9 p^{2} T^{5} + p^{3} T^{6} \) |
| 83 | $A_4\times C_2$ | \( 1 + 16 T + 304 T^{2} + 2613 T^{3} + 304 p T^{4} + 16 p^{2} T^{5} + p^{3} T^{6} \) |
| 89 | $A_4\times C_2$ | \( 1 - 5 T + 259 T^{2} - 891 T^{3} + 259 p T^{4} - 5 p^{2} T^{5} + p^{3} T^{6} \) |
| 97 | $A_4\times C_2$ | \( 1 + 5 T + 122 T^{2} - 219 T^{3} + 122 p T^{4} + 5 p^{2} T^{5} + p^{3} T^{6} \) |
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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
−7.46836246232969281859259778709, −7.15499357234526488646541123102, −6.88366889187667061264006806164, −6.63807028833746263630585631779, −6.14100759199415691748954335292, −6.14029808428758175069274343325, −5.73760790204501815927437412603, −5.48603547886983487259632495454, −5.39453012248102383284089808500, −4.96711698134447443500017922298, −4.83218458492577701461055795075, −4.83159231670333882456599014597, −4.55014710503690894056304793394, −4.16655739806712117005005788203, −4.09647229806733100431844738850, −4.02310379444992843536757895038, −3.23471640339378557341973275944, −3.18866982013216305274387279240, −3.13246078384785048990267992256, −2.14690901412227149522107149218, −2.08301381156695593002848589701, −1.85340977498473678510209724924, −1.43487749347466136276338454014, −1.16178119709563314028245008067, −1.15972287209929871956668103191, 0, 0, 0,
1.15972287209929871956668103191, 1.16178119709563314028245008067, 1.43487749347466136276338454014, 1.85340977498473678510209724924, 2.08301381156695593002848589701, 2.14690901412227149522107149218, 3.13246078384785048990267992256, 3.18866982013216305274387279240, 3.23471640339378557341973275944, 4.02310379444992843536757895038, 4.09647229806733100431844738850, 4.16655739806712117005005788203, 4.55014710503690894056304793394, 4.83159231670333882456599014597, 4.83218458492577701461055795075, 4.96711698134447443500017922298, 5.39453012248102383284089808500, 5.48603547886983487259632495454, 5.73760790204501815927437412603, 6.14029808428758175069274343325, 6.14100759199415691748954335292, 6.63807028833746263630585631779, 6.88366889187667061264006806164, 7.15499357234526488646541123102, 7.46836246232969281859259778709
