| L(s) = 1 | − 3·2-s + 6·4-s + 6·5-s − 3·7-s − 10·8-s − 18·10-s + 12·11-s + 9·14-s + 15·16-s + 12·17-s + 36·20-s − 36·22-s + 12·23-s + 12·25-s − 18·28-s + 6·29-s − 3·31-s − 21·32-s − 36·34-s − 18·35-s + 3·37-s − 60·40-s + 9·41-s − 3·43-s + 72·44-s − 36·46-s + 15·47-s + ⋯ |
| L(s) = 1 | − 2.12·2-s + 3·4-s + 2.68·5-s − 1.13·7-s − 3.53·8-s − 5.69·10-s + 3.61·11-s + 2.40·14-s + 15/4·16-s + 2.91·17-s + 8.04·20-s − 7.67·22-s + 2.50·23-s + 12/5·25-s − 3.40·28-s + 1.11·29-s − 0.538·31-s − 3.71·32-s − 6.17·34-s − 3.04·35-s + 0.493·37-s − 9.48·40-s + 1.40·41-s − 0.457·43-s + 10.8·44-s − 5.30·46-s + 2.18·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{3} \cdot 3^{6} \cdot 19^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{3} \cdot 3^{6} \cdot 19^{6}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(6.891043289\) |
| \(L(\frac12)\) |
\(\approx\) |
\(6.891043289\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 19 | | \( 1 \) |
| good | 5 | $A_4\times C_2$ | \( 1 - 6 T + 24 T^{2} - 63 T^{3} + 24 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 7 | $A_4\times C_2$ | \( 1 + 3 T + 15 T^{2} + 43 T^{3} + 15 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} \) |
| 11 | $A_4\times C_2$ | \( 1 - 12 T + 78 T^{2} - 315 T^{3} + 78 p T^{4} - 12 p^{2} T^{5} + p^{3} T^{6} \) |
| 13 | $C_6$ | \( 1 + 19 T^{3} + p^{3} T^{6} \) |
| 17 | $A_4\times C_2$ | \( 1 - 12 T + 96 T^{2} - 27 p T^{3} + 96 p T^{4} - 12 p^{2} T^{5} + p^{3} T^{6} \) |
| 23 | $A_4\times C_2$ | \( 1 - 12 T + 96 T^{2} - 549 T^{3} + 96 p T^{4} - 12 p^{2} T^{5} + p^{3} T^{6} \) |
| 29 | $A_4\times C_2$ | \( 1 - 6 T + 60 T^{2} - 297 T^{3} + 60 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 31 | $A_4\times C_2$ | \( 1 + 3 T + 87 T^{2} + 169 T^{3} + 87 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} \) |
| 37 | $A_4\times C_2$ | \( 1 - 3 T + 102 T^{2} - 203 T^{3} + 102 p T^{4} - 3 p^{2} T^{5} + p^{3} T^{6} \) |
| 41 | $A_4\times C_2$ | \( 1 - 9 T + 3 p T^{2} - 711 T^{3} + 3 p^{2} T^{4} - 9 p^{2} T^{5} + p^{3} T^{6} \) |
| 43 | $A_4\times C_2$ | \( 1 + 3 T + 105 T^{2} + 259 T^{3} + 105 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} \) |
| 47 | $A_4\times C_2$ | \( 1 - 15 T + 177 T^{2} - 1251 T^{3} + 177 p T^{4} - 15 p^{2} T^{5} + p^{3} T^{6} \) |
| 53 | $A_4\times C_2$ | \( 1 - 3 T + 78 T^{2} - 99 T^{3} + 78 p T^{4} - 3 p^{2} T^{5} + p^{3} T^{6} \) |
| 59 | $A_4\times C_2$ | \( 1 + 60 T^{2} - 153 T^{3} + 60 p T^{4} + p^{3} T^{6} \) |
| 61 | $A_4\times C_2$ | \( 1 + 3 T + 159 T^{2} + 367 T^{3} + 159 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} \) |
| 67 | $A_4\times C_2$ | \( 1 + 6 T + 174 T^{2} + 715 T^{3} + 174 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 71 | $A_4\times C_2$ | \( 1 - 9 T + 3 p T^{2} - 1197 T^{3} + 3 p^{2} T^{4} - 9 p^{2} T^{5} + p^{3} T^{6} \) |
| 73 | $A_4\times C_2$ | \( 1 - 3 T - 15 T^{2} + 301 T^{3} - 15 p T^{4} - 3 p^{2} T^{5} + p^{3} T^{6} \) |
| 79 | $A_4\times C_2$ | \( 1 - 9 T + 189 T^{2} - 1349 T^{3} + 189 p T^{4} - 9 p^{2} T^{5} + p^{3} T^{6} \) |
| 83 | $A_4\times C_2$ | \( 1 + 3 T - 21 T^{2} - 1395 T^{3} - 21 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} \) |
| 89 | $A_4\times C_2$ | \( 1 + 3 T - 3 T^{2} - 1359 T^{3} - 3 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} \) |
| 97 | $A_4\times C_2$ | \( 1 + 12 T + 168 T^{2} + 1051 T^{3} + 168 p T^{4} + 12 p^{2} T^{5} + p^{3} T^{6} \) |
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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
−7.18340173852895775166647459184, −6.67200390173642659402469057806, −6.65729032568539623783758729465, −6.56967489867915630685908069986, −6.12604398191871754500199868082, −6.00316274934175470935553854044, −5.99618761540178188030028491908, −5.56120878133479800760609609104, −5.32491431216083051557335746127, −5.29135534257906071978196490358, −4.65990066300126963713787813504, −4.41268594241361680733764351887, −3.93410414657193536248639751905, −3.66050662567384534397101073063, −3.51001119133248703680405803748, −3.22957705973311716778582110204, −2.80478775579980634938781102664, −2.63978588312758913252919699431, −2.42107278226081138177545812138, −1.84809272519518223902181234496, −1.52278260691966232216219358073, −1.46556929044452028457354594508, −1.16123802469907590597379440016, −0.77498690164416021422554000529, −0.72870721255598761507121966676,
0.72870721255598761507121966676, 0.77498690164416021422554000529, 1.16123802469907590597379440016, 1.46556929044452028457354594508, 1.52278260691966232216219358073, 1.84809272519518223902181234496, 2.42107278226081138177545812138, 2.63978588312758913252919699431, 2.80478775579980634938781102664, 3.22957705973311716778582110204, 3.51001119133248703680405803748, 3.66050662567384534397101073063, 3.93410414657193536248639751905, 4.41268594241361680733764351887, 4.65990066300126963713787813504, 5.29135534257906071978196490358, 5.32491431216083051557335746127, 5.56120878133479800760609609104, 5.99618761540178188030028491908, 6.00316274934175470935553854044, 6.12604398191871754500199868082, 6.56967489867915630685908069986, 6.65729032568539623783758729465, 6.67200390173642659402469057806, 7.18340173852895775166647459184
