| L(s) = 1 | − 2·2-s − 3·3-s + 2·4-s + 6·6-s + 3·7-s − 8-s + 6·9-s − 3·11-s − 6·12-s + 4·13-s − 6·14-s − 4·16-s − 8·17-s − 12·18-s − 8·19-s − 9·21-s + 6·22-s − 10·23-s + 3·24-s − 8·26-s − 10·27-s + 6·28-s − 4·29-s − 2·31-s + 8·32-s + 9·33-s + 16·34-s + ⋯ |
| L(s) = 1 | − 1.41·2-s − 1.73·3-s + 4-s + 2.44·6-s + 1.13·7-s − 0.353·8-s + 2·9-s − 0.904·11-s − 1.73·12-s + 1.10·13-s − 1.60·14-s − 16-s − 1.94·17-s − 2.82·18-s − 1.83·19-s − 1.96·21-s + 1.27·22-s − 2.08·23-s + 0.612·24-s − 1.56·26-s − 1.92·27-s + 1.13·28-s − 0.742·29-s − 0.359·31-s + 1.41·32-s + 1.56·33-s + 2.74·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{3} \cdot 5^{6} \cdot 7^{3} \cdot 11^{3}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{3} \cdot 5^{6} \cdot 7^{3} \cdot 11^{3}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.6681214041\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6681214041\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 + T )^{3} \) |
| 5 | | \( 1 \) |
| 7 | $C_1$ | \( ( 1 - T )^{3} \) |
| 11 | $C_1$ | \( ( 1 + T )^{3} \) |
| good | 2 | $S_4\times C_2$ | \( 1 + p T + p T^{2} + T^{3} + p^{2} T^{4} + p^{3} T^{5} + p^{3} T^{6} \) |
| 13 | $S_4\times C_2$ | \( 1 - 4 T + 12 T^{2} - 10 T^{3} + 12 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} \) |
| 17 | $S_4\times C_2$ | \( 1 + 8 T + 11 T^{2} - 56 T^{3} + 11 p T^{4} + 8 p^{2} T^{5} + p^{3} T^{6} \) |
| 19 | $S_4\times C_2$ | \( 1 + 8 T + 72 T^{2} + 308 T^{3} + 72 p T^{4} + 8 p^{2} T^{5} + p^{3} T^{6} \) |
| 23 | $S_4\times C_2$ | \( 1 + 10 T + 81 T^{2} + 396 T^{3} + 81 p T^{4} + 10 p^{2} T^{5} + p^{3} T^{6} \) |
| 29 | $S_4\times C_2$ | \( 1 + 4 T + 60 T^{2} + 138 T^{3} + 60 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} \) |
| 31 | $S_4\times C_2$ | \( 1 + 2 T + 17 T^{2} - 132 T^{3} + 17 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 37 | $S_4\times C_2$ | \( 1 + 68 T^{2} - 106 T^{3} + 68 p T^{4} + p^{3} T^{6} \) |
| 41 | $S_4\times C_2$ | \( 1 - 14 T + 163 T^{2} - 1116 T^{3} + 163 p T^{4} - 14 p^{2} T^{5} + p^{3} T^{6} \) |
| 43 | $S_4\times C_2$ | \( 1 - 14 T + 85 T^{2} - 356 T^{3} + 85 p T^{4} - 14 p^{2} T^{5} + p^{3} T^{6} \) |
| 47 | $S_4\times C_2$ | \( 1 + 80 T^{2} - 32 T^{3} + 80 p T^{4} + p^{3} T^{6} \) |
| 53 | $S_4\times C_2$ | \( 1 + 143 T^{2} - 8 T^{3} + 143 p T^{4} + p^{3} T^{6} \) |
| 59 | $S_4\times C_2$ | \( 1 + 120 T^{2} - 52 T^{3} + 120 p T^{4} + p^{3} T^{6} \) |
| 61 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{3} \) |
| 67 | $S_4\times C_2$ | \( 1 - 4 T + 116 T^{2} - 300 T^{3} + 116 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} \) |
| 71 | $S_4\times C_2$ | \( 1 + 12 T + 197 T^{2} + 1448 T^{3} + 197 p T^{4} + 12 p^{2} T^{5} + p^{3} T^{6} \) |
| 73 | $S_4\times C_2$ | \( 1 - 20 T + 320 T^{2} - 3054 T^{3} + 320 p T^{4} - 20 p^{2} T^{5} + p^{3} T^{6} \) |
| 79 | $S_4\times C_2$ | \( 1 - 12 T + 221 T^{2} - 1640 T^{3} + 221 p T^{4} - 12 p^{2} T^{5} + p^{3} T^{6} \) |
| 83 | $S_4\times C_2$ | \( 1 + 6 T + 117 T^{2} + 500 T^{3} + 117 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 89 | $S_4\times C_2$ | \( 1 - 26 T + 407 T^{2} - 4300 T^{3} + 407 p T^{4} - 26 p^{2} T^{5} + p^{3} T^{6} \) |
| 97 | $S_4\times C_2$ | \( 1 - 4 T + 171 T^{2} - 544 T^{3} + 171 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} \) |
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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
−7.14658130778753100978230606502, −6.96810165013271856414379147064, −6.72215647399081415707391310241, −6.34550109908446233044778623116, −6.22163916799176645262294683060, −6.04543902121399297692107561192, −5.97192293392954886811664323102, −5.55389503331399996785643840235, −5.22056707018586083229444120762, −5.13560371071555989848359884992, −4.60907806443572559478852371693, −4.43176956072859133841734098150, −4.33709920639139925394699040642, −4.14406062493987152710783444959, −3.78828461773882395978268917429, −3.59385735051630310870946507920, −2.89901995771625584227846946756, −2.44782139266176723191044398955, −2.19786281323460252460187157105, −2.13035033377331299753982991456, −1.71872178744974587639122314576, −1.60859633750760873327966472775, −0.790959217081437585732768618547, −0.46212839077164552854046041377, −0.45260080174658908881155098716,
0.45260080174658908881155098716, 0.46212839077164552854046041377, 0.790959217081437585732768618547, 1.60859633750760873327966472775, 1.71872178744974587639122314576, 2.13035033377331299753982991456, 2.19786281323460252460187157105, 2.44782139266176723191044398955, 2.89901995771625584227846946756, 3.59385735051630310870946507920, 3.78828461773882395978268917429, 4.14406062493987152710783444959, 4.33709920639139925394699040642, 4.43176956072859133841734098150, 4.60907806443572559478852371693, 5.13560371071555989848359884992, 5.22056707018586083229444120762, 5.55389503331399996785643840235, 5.97192293392954886811664323102, 6.04543902121399297692107561192, 6.22163916799176645262294683060, 6.34550109908446233044778623116, 6.72215647399081415707391310241, 6.96810165013271856414379147064, 7.14658130778753100978230606502
