| L(s) = 1 | + 3·3-s − 3·7-s + 6·9-s − 4·11-s − 6·13-s + 4·17-s − 3·19-s − 9·21-s − 4·23-s − 15·25-s + 10·27-s − 12·29-s + 4·31-s − 12·33-s − 10·37-s − 18·39-s − 2·41-s − 4·43-s − 2·47-s + 6·49-s + 12·51-s − 24·53-s − 9·57-s + 4·59-s − 14·61-s − 18·63-s − 12·69-s + ⋯ |
| L(s) = 1 | + 1.73·3-s − 1.13·7-s + 2·9-s − 1.20·11-s − 1.66·13-s + 0.970·17-s − 0.688·19-s − 1.96·21-s − 0.834·23-s − 3·25-s + 1.92·27-s − 2.22·29-s + 0.718·31-s − 2.08·33-s − 1.64·37-s − 2.88·39-s − 0.312·41-s − 0.609·43-s − 0.291·47-s + 6/7·49-s + 1.68·51-s − 3.29·53-s − 1.19·57-s + 0.520·59-s − 1.79·61-s − 2.26·63-s − 1.44·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{9} \cdot 3^{3} \cdot 7^{3} \cdot 19^{3}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & -\,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{9} \cdot 3^{3} \cdot 7^{3} \cdot 19^{3}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr=\mathstrut & -\,\Lambda(1-s)\end{aligned}\]
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 | | \( 1 \) |
| 3 | $C_1$ | \( ( 1 - T )^{3} \) |
| 7 | $C_1$ | \( ( 1 + T )^{3} \) |
| 19 | $C_1$ | \( ( 1 + T )^{3} \) |
| good | 5 | $C_2$ | \( ( 1 + p T^{2} )^{3} \) |
| 11 | $S_4\times C_2$ | \( 1 + 4 T + 21 T^{2} + 72 T^{3} + 21 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} \) |
| 13 | $S_4\times C_2$ | \( 1 + 6 T + 23 T^{2} + 92 T^{3} + 23 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 17 | $S_4\times C_2$ | \( 1 - 4 T + 27 T^{2} - 72 T^{3} + 27 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} \) |
| 23 | $S_4\times C_2$ | \( 1 + 4 T + 9 T^{2} - 72 T^{3} + 9 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} \) |
| 29 | $S_4\times C_2$ | \( 1 + 12 T + 107 T^{2} + 664 T^{3} + 107 p T^{4} + 12 p^{2} T^{5} + p^{3} T^{6} \) |
| 31 | $S_4\times C_2$ | \( 1 - 4 T + 37 T^{2} + 8 T^{3} + 37 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} \) |
| 37 | $S_4\times C_2$ | \( 1 + 10 T + 83 T^{2} + 396 T^{3} + 83 p T^{4} + 10 p^{2} T^{5} + p^{3} T^{6} \) |
| 41 | $S_4\times C_2$ | \( 1 + 2 T + 55 T^{2} + 156 T^{3} + 55 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 43 | $S_4\times C_2$ | \( 1 + 4 T + 65 T^{2} + 216 T^{3} + 65 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} \) |
| 47 | $S_4\times C_2$ | \( 1 + 2 T + 125 T^{2} + 172 T^{3} + 125 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 53 | $S_4\times C_2$ | \( 1 + 24 T + 323 T^{2} + 2816 T^{3} + 323 p T^{4} + 24 p^{2} T^{5} + p^{3} T^{6} \) |
| 59 | $S_4\times C_2$ | \( 1 - 4 T + 65 T^{2} - 536 T^{3} + 65 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} \) |
| 61 | $S_4\times C_2$ | \( 1 + 14 T + 131 T^{2} + 1044 T^{3} + 131 p T^{4} + 14 p^{2} T^{5} + p^{3} T^{6} \) |
| 67 | $D_{6}$ | \( 1 + 45 T^{2} + 128 T^{3} + 45 p T^{4} + p^{3} T^{6} \) |
| 71 | $S_4\times C_2$ | \( 1 + 14 T + 3 p T^{2} + 1956 T^{3} + 3 p^{2} T^{4} + 14 p^{2} T^{5} + p^{3} T^{6} \) |
| 73 | $S_4\times C_2$ | \( 1 - 2 T + 151 T^{2} - 284 T^{3} + 151 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 79 | $S_4\times C_2$ | \( 1 - 4 T + 113 T^{2} - 984 T^{3} + 113 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} \) |
| 83 | $S_4\times C_2$ | \( 1 + 14 T + 185 T^{2} + 1300 T^{3} + 185 p T^{4} + 14 p^{2} T^{5} + p^{3} T^{6} \) |
| 89 | $S_4\times C_2$ | \( 1 + 18 T + 263 T^{2} + 2620 T^{3} + 263 p T^{4} + 18 p^{2} T^{5} + p^{3} T^{6} \) |
| 97 | $S_4\times C_2$ | \( 1 + 2 T + 195 T^{2} + 692 T^{3} + 195 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \) |
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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
−8.109445063262521776567204048584, −7.68378547798668239242355589224, −7.67378433711953986256666906695, −7.59861137563680160646433092771, −7.08675817711983935797107690471, −6.88293328983520727975241193468, −6.77670272815470916751001441231, −6.11924444577045419169008902863, −6.11561959575964186374835783784, −5.88781429319536879421774114884, −5.28853088257154114054014988122, −5.24247050852249481085549242203, −5.16266868759834797226565001717, −4.44726954790598706654071024306, −4.28957533020980283845120205006, −4.06068804146089013802646969735, −3.70443715222408585438457468847, −3.51266154468211163360502961187, −3.14028059375825214603687456599, −2.87256943672115723737366308386, −2.55503576553075487499078067936, −2.49179577074201826093883513892, −1.70592124437086147635292832874, −1.68510107726780480065162681910, −1.60213031466138144721619213968, 0, 0, 0,
1.60213031466138144721619213968, 1.68510107726780480065162681910, 1.70592124437086147635292832874, 2.49179577074201826093883513892, 2.55503576553075487499078067936, 2.87256943672115723737366308386, 3.14028059375825214603687456599, 3.51266154468211163360502961187, 3.70443715222408585438457468847, 4.06068804146089013802646969735, 4.28957533020980283845120205006, 4.44726954790598706654071024306, 5.16266868759834797226565001717, 5.24247050852249481085549242203, 5.28853088257154114054014988122, 5.88781429319536879421774114884, 6.11561959575964186374835783784, 6.11924444577045419169008902863, 6.77670272815470916751001441231, 6.88293328983520727975241193468, 7.08675817711983935797107690471, 7.59861137563680160646433092771, 7.67378433711953986256666906695, 7.68378547798668239242355589224, 8.109445063262521776567204048584
