| L(s) = 1 | + 2-s − 3·4-s + 4·5-s − 10·7-s − 4·8-s + 4·10-s − 11-s − 10·14-s + 3·16-s + 7·17-s − 11·19-s − 12·20-s − 22-s − 2·23-s − 2·25-s + 30·28-s + 8·29-s − 8·31-s + 6·32-s + 7·34-s − 40·35-s − 14·37-s − 11·38-s − 16·40-s + 41-s − 3·43-s + 3·44-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 3/2·4-s + 1.78·5-s − 3.77·7-s − 1.41·8-s + 1.26·10-s − 0.301·11-s − 2.67·14-s + 3/4·16-s + 1.69·17-s − 2.52·19-s − 2.68·20-s − 0.213·22-s − 0.417·23-s − 2/5·25-s + 5.66·28-s + 1.48·29-s − 1.43·31-s + 1.06·32-s + 1.20·34-s − 6.76·35-s − 2.30·37-s − 1.78·38-s − 2.52·40-s + 0.156·41-s − 0.457·43-s + 0.452·44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{6} \cdot 13^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & -\,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{6} \cdot 13^{6}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr=\mathstrut & -\,\Lambda(1-s)\end{aligned}\]
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 | 3 | | \( 1 \) |
| 13 | | \( 1 \) |
| good | 2 | $A_4\times C_2$ | \( 1 - T + p^{2} T^{2} - 3 T^{3} + p^{3} T^{4} - p^{2} T^{5} + p^{3} T^{6} \) |
| 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
−9.075095779072428104383569832688, −8.563137452600505583038366475641, −8.423757190499622701131216757011, −8.344909557397332682144848278369, −7.51418938028855006725284722294, −7.42971359446796852427066526973, −7.06393085823028855549081203275, −6.77676663227556136629680344727, −6.32549587139015371362437589084, −6.18428676484567164978347077363, −6.09284928634346219876523376365, −5.79726859550645205918331966796, −5.72970142266310747035770052539, −5.18318281863811362304794312606, −4.93500532239383697754949419236, −4.64982782161842097768221998969, −3.98882603353342360475565429560, −3.95115751257984579506791085647, −3.72021861383773522017958466972, −3.29718836709218914119442886516, −2.95547721841344800016113278909, −2.78142834641264025903654852269, −2.35155503429309620116442076996, −1.63451042249254106473523472232, −1.48025328363071115284434145052, 0, 0, 0,
1.48025328363071115284434145052, 1.63451042249254106473523472232, 2.35155503429309620116442076996, 2.78142834641264025903654852269, 2.95547721841344800016113278909, 3.29718836709218914119442886516, 3.72021861383773522017958466972, 3.95115751257984579506791085647, 3.98882603353342360475565429560, 4.64982782161842097768221998969, 4.93500532239383697754949419236, 5.18318281863811362304794312606, 5.72970142266310747035770052539, 5.79726859550645205918331966796, 6.09284928634346219876523376365, 6.18428676484567164978347077363, 6.32549587139015371362437589084, 6.77676663227556136629680344727, 7.06393085823028855549081203275, 7.42971359446796852427066526973, 7.51418938028855006725284722294, 8.344909557397332682144848278369, 8.423757190499622701131216757011, 8.563137452600505583038366475641, 9.075095779072428104383569832688
