| L(s) = 1 | + 3·2-s + 3·4-s + 6·5-s − 3·7-s + 18·10-s − 3·13-s − 9·14-s − 3·16-s + 18·17-s + 3·19-s + 18·20-s + 9·23-s + 12·25-s − 9·26-s − 9·28-s − 3·29-s + 12·31-s − 6·32-s + 54·34-s − 18·35-s + 9·38-s + 12·41-s − 9·43-s + 27·46-s + 3·47-s − 12·49-s + 36·50-s + ⋯ |
| L(s) = 1 | + 2.12·2-s + 3/2·4-s + 2.68·5-s − 1.13·7-s + 5.69·10-s − 0.832·13-s − 2.40·14-s − 3/4·16-s + 4.36·17-s + 0.688·19-s + 4.02·20-s + 1.87·23-s + 12/5·25-s − 1.76·26-s − 1.70·28-s − 0.557·29-s + 2.15·31-s − 1.06·32-s + 9.26·34-s − 3.04·35-s + 1.45·38-s + 1.87·41-s − 1.37·43-s + 3.98·46-s + 0.437·47-s − 1.71·49-s + 5.09·50-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{6} \cdot 127^{3}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{6} \cdot 127^{3}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(15.44973575\) |
| \(L(\frac12)\) |
\(\approx\) |
\(15.44973575\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 127 | $C_1$ | \( ( 1 + T )^{3} \) |
| good | 2 | $A_4\times C_2$ | \( 1 - 3 T + 3 p T^{2} - 9 T^{3} + 3 p^{2} T^{4} - 3 p^{2} T^{5} + p^{3} T^{6} \) |
| 5 | $A_4\times C_2$ | \( 1 - 6 T + 24 T^{2} - 61 T^{3} + 24 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 7 | $A_4\times C_2$ | \( 1 + 3 T + 3 p T^{2} + 39 T^{3} + 3 p^{2} T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} \) |
| 11 | $A_4\times C_2$ | \( 1 + 12 T^{2} + 37 T^{3} + 12 p T^{4} + p^{3} T^{6} \) |
| 13 | $A_4\times C_2$ | \( 1 + 3 T + 21 T^{2} + 41 T^{3} + 21 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} \) |
| 17 | $A_4\times C_2$ | \( 1 - 18 T + 156 T^{2} - 811 T^{3} + 156 p T^{4} - 18 p^{2} T^{5} + p^{3} T^{6} \) |
| 19 | $A_4\times C_2$ | \( 1 - 3 T + 3 p T^{2} - 113 T^{3} + 3 p^{2} T^{4} - 3 p^{2} T^{5} + p^{3} T^{6} \) |
| 23 | $A_4\times C_2$ | \( 1 - 9 T + 87 T^{2} - 405 T^{3} + 87 p T^{4} - 9 p^{2} T^{5} + p^{3} T^{6} \) |
| 29 | $A_4\times C_2$ | \( 1 + 3 T + 69 T^{2} + 171 T^{3} + 69 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} \) |
| 31 | $A_4\times C_2$ | \( 1 - 12 T + 120 T^{2} - 761 T^{3} + 120 p T^{4} - 12 p^{2} T^{5} + p^{3} T^{6} \) |
| 37 | $A_4\times C_2$ | \( 1 + 27 T^{2} + 8 p T^{3} + 27 p T^{4} + p^{3} T^{6} \) |
| 41 | $A_4\times C_2$ | \( 1 - 12 T + 3 p T^{2} - 792 T^{3} + 3 p^{2} T^{4} - 12 p^{2} T^{5} + p^{3} T^{6} \) |
| 43 | $A_4\times C_2$ | \( 1 + 9 T + 48 T^{2} + 261 T^{3} + 48 p T^{4} + 9 p^{2} T^{5} + p^{3} T^{6} \) |
| 47 | $A_4\times C_2$ | \( 1 - 3 T + 60 T^{2} + 97 T^{3} + 60 p T^{4} - 3 p^{2} T^{5} + p^{3} T^{6} \) |
| 53 | $A_4\times C_2$ | \( 1 + 3 T + 33 T^{2} + 261 T^{3} + 33 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} \) |
| 59 | $A_4\times C_2$ | \( 1 + 156 T^{2} - 37 T^{3} + 156 p T^{4} + p^{3} T^{6} \) |
| 61 | $A_4\times C_2$ | \( 1 + 3 T + 30 T^{2} + 59 T^{3} + 30 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} \) |
| 67 | $A_4\times C_2$ | \( 1 + 3 T + 3 p T^{2} + 401 T^{3} + 3 p^{2} T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} \) |
| 71 | $A_4\times C_2$ | \( 1 + 3 T + 60 T^{2} - 441 T^{3} + 60 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} \) |
| 73 | $A_4\times C_2$ | \( 1 - 3 T + 105 T^{2} - 169 T^{3} + 105 p T^{4} - 3 p^{2} T^{5} + p^{3} T^{6} \) |
| 79 | $A_4\times C_2$ | \( 1 - 9 T + 117 T^{2} - 1351 T^{3} + 117 p T^{4} - 9 p^{2} T^{5} + p^{3} T^{6} \) |
| 83 | $A_4\times C_2$ | \( 1 + 12 T + 24 T^{2} - 657 T^{3} + 24 p T^{4} + 12 p^{2} T^{5} + p^{3} T^{6} \) |
| 89 | $A_4\times C_2$ | \( 1 - 33 T + 573 T^{2} - 6471 T^{3} + 573 p T^{4} - 33 p^{2} T^{5} + p^{3} T^{6} \) |
| 97 | $A_4\times C_2$ | \( 1 + 15 T + 285 T^{2} + 2873 T^{3} + 285 p T^{4} + 15 p^{2} T^{5} + p^{3} T^{6} \) |
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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
−9.125983963949881766048969892689, −8.396502150612938005985198821816, −8.050945162835337185163066202756, −7.70867974047228989910145295564, −7.58516136863581889998845578290, −7.25465095390858026481375133991, −6.81897748995771082853049540358, −6.45757487143997608506924862902, −6.12403464313831684740738313643, −6.03178540787566895065379386069, −5.83813426399427226935726206777, −5.37355734183641939335254487774, −5.35876606303302228946362368878, −5.00193953733705428631171851770, −4.69008902325513668457534153049, −4.66861037648709474466685992652, −3.72097265799901943965579481945, −3.56498678266320906164869455070, −3.41541258122782416791013413117, −2.86025707375385917019174980777, −2.81497325772301451659406931914, −2.35739242152306823194931347702, −1.66216969591739152246595353089, −1.28841828421406454033934078856, −0.858282475861201521512282086873,
0.858282475861201521512282086873, 1.28841828421406454033934078856, 1.66216969591739152246595353089, 2.35739242152306823194931347702, 2.81497325772301451659406931914, 2.86025707375385917019174980777, 3.41541258122782416791013413117, 3.56498678266320906164869455070, 3.72097265799901943965579481945, 4.66861037648709474466685992652, 4.69008902325513668457534153049, 5.00193953733705428631171851770, 5.35876606303302228946362368878, 5.37355734183641939335254487774, 5.83813426399427226935726206777, 6.03178540787566895065379386069, 6.12403464313831684740738313643, 6.45757487143997608506924862902, 6.81897748995771082853049540358, 7.25465095390858026481375133991, 7.58516136863581889998845578290, 7.70867974047228989910145295564, 8.050945162835337185163066202756, 8.396502150612938005985198821816, 9.125983963949881766048969892689
