| L(s) = 1 | − 3-s + 4·11-s − 8·13-s − 10·17-s − 2·19-s + 3·23-s − 2·27-s − 3·31-s − 4·33-s − 6·37-s + 8·39-s + 11·41-s + 11·43-s − 4·47-s + 10·51-s − 14·53-s + 2·57-s − 5·59-s + 17·61-s − 20·67-s − 3·69-s − 19·71-s − 12·73-s − 79-s + 2·81-s − 28·83-s − 14·89-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 1.20·11-s − 2.21·13-s − 2.42·17-s − 0.458·19-s + 0.625·23-s − 0.384·27-s − 0.538·31-s − 0.696·33-s − 0.986·37-s + 1.28·39-s + 1.71·41-s + 1.67·43-s − 0.583·47-s + 1.40·51-s − 1.92·53-s + 0.264·57-s − 0.650·59-s + 2.17·61-s − 2.44·67-s − 0.361·69-s − 2.25·71-s − 1.40·73-s − 0.112·79-s + 2/9·81-s − 3.07·83-s − 1.48·89-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{6} \cdot 5^{6} \cdot 7^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & -\,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{6} \cdot 5^{6} \cdot 7^{6}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr=\mathstrut & -\,\Lambda(1-s)\end{aligned}\]
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 | 2 | | \( 1 \) |
| 5 | | \( 1 \) |
| 7 | | \( 1 \) |
| good | 3 | $S_4\times C_2$ | \( 1 + T + T^{2} + p T^{3} + p T^{4} + p^{2} T^{5} + p^{3} T^{6} \) |
| 11 | $S_4\times C_2$ | \( 1 - 4 T + 30 T^{2} - 79 T^{3} + 30 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} \) |
| 13 | $S_4\times C_2$ | \( 1 + 8 T + 4 p T^{2} + 205 T^{3} + 4 p^{2} T^{4} + 8 p^{2} T^{5} + p^{3} T^{6} \) |
| 17 | $S_4\times C_2$ | \( 1 + 10 T + 60 T^{2} + 259 T^{3} + 60 p T^{4} + 10 p^{2} T^{5} + p^{3} T^{6} \) |
| 19 | $S_4\times C_2$ | \( 1 + 2 T + 34 T^{2} + 97 T^{3} + 34 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 23 | $S_4\times C_2$ | \( 1 - 3 T + 33 T^{2} - 57 T^{3} + 33 p T^{4} - 3 p^{2} T^{5} + p^{3} T^{6} \) |
| 29 | $S_4\times C_2$ | \( 1 + 42 T^{2} + 81 T^{3} + 42 p T^{4} + p^{3} T^{6} \) |
| 31 | $S_4\times C_2$ | \( 1 + 3 T + 51 T^{2} + 223 T^{3} + 51 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} \) |
| 37 | $S_4\times C_2$ | \( 1 + 6 T + 48 T^{2} + 295 T^{3} + 48 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 41 | $S_4\times C_2$ | \( 1 - 11 T + 45 T^{2} - 29 T^{3} + 45 p T^{4} - 11 p^{2} T^{5} + p^{3} T^{6} \) |
| 43 | $S_4\times C_2$ | \( 1 - 11 T + 143 T^{2} - 875 T^{3} + 143 p T^{4} - 11 p^{2} T^{5} + p^{3} T^{6} \) |
| 47 | $S_4\times C_2$ | \( 1 + 4 T + 120 T^{2} + 367 T^{3} + 120 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} \) |
| 53 | $S_4\times C_2$ | \( 1 + 14 T + 144 T^{2} + 935 T^{3} + 144 p T^{4} + 14 p^{2} T^{5} + p^{3} T^{6} \) |
| 59 | $S_4\times C_2$ | \( 1 + 5 T + 3 p T^{2} + 581 T^{3} + 3 p^{2} T^{4} + 5 p^{2} T^{5} + p^{3} T^{6} \) |
| 61 | $S_4\times C_2$ | \( 1 - 17 T + 209 T^{2} - 2075 T^{3} + 209 p T^{4} - 17 p^{2} T^{5} + p^{3} T^{6} \) |
| 67 | $S_4\times C_2$ | \( 1 + 20 T + 301 T^{2} + 2752 T^{3} + 301 p T^{4} + 20 p^{2} T^{5} + p^{3} T^{6} \) |
| 71 | $S_4\times C_2$ | \( 1 + 19 T + 300 T^{2} + 2743 T^{3} + 300 p T^{4} + 19 p^{2} T^{5} + p^{3} T^{6} \) |
| 73 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{3} \) |
| 79 | $S_4\times C_2$ | \( 1 + T + 119 T^{2} + 607 T^{3} + 119 p T^{4} + p^{2} T^{5} + p^{3} T^{6} \) |
| 83 | $S_4\times C_2$ | \( 1 + 28 T + 378 T^{2} + 3667 T^{3} + 378 p T^{4} + 28 p^{2} T^{5} + p^{3} T^{6} \) |
| 89 | $S_4\times C_2$ | \( 1 + 14 T + 306 T^{2} + 2447 T^{3} + 306 p T^{4} + 14 p^{2} T^{5} + p^{3} T^{6} \) |
| 97 | $S_4\times C_2$ | \( 1 + 15 T + 273 T^{2} + 2905 T^{3} + 273 p T^{4} + 15 p^{2} T^{5} + p^{3} T^{6} \) |
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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
−7.38326589098707973131463622273, −7.37846148349129558650874429613, −7.15933245211294192484394288746, −7.06486407701617631813353106427, −6.72621762316305402496610937318, −6.45015244278290398413794175758, −6.30564278188283802954254347446, −5.79320526326866992594591918765, −5.77819942504301417483198922803, −5.68518648548065216559944735124, −5.10519408038735440619761125044, −4.90390435291548281691570260909, −4.66807940086336215300358321767, −4.32770196285317151248163778856, −4.24805233955196192141229409764, −4.17507797170663660134930533199, −3.72611543418915920743764371028, −3.12780180591418807188250946158, −3.04021801066998193085758858297, −2.57447042441233340321432350054, −2.52109966762684419845790942782, −2.05180450179778004754934583423, −1.79735645751241287585330547448, −1.28838503923029242039403241500, −1.13338499387032574266983770810, 0, 0, 0,
1.13338499387032574266983770810, 1.28838503923029242039403241500, 1.79735645751241287585330547448, 2.05180450179778004754934583423, 2.52109966762684419845790942782, 2.57447042441233340321432350054, 3.04021801066998193085758858297, 3.12780180591418807188250946158, 3.72611543418915920743764371028, 4.17507797170663660134930533199, 4.24805233955196192141229409764, 4.32770196285317151248163778856, 4.66807940086336215300358321767, 4.90390435291548281691570260909, 5.10519408038735440619761125044, 5.68518648548065216559944735124, 5.77819942504301417483198922803, 5.79320526326866992594591918765, 6.30564278188283802954254347446, 6.45015244278290398413794175758, 6.72621762316305402496610937318, 7.06486407701617631813353106427, 7.15933245211294192484394288746, 7.37846148349129558650874429613, 7.38326589098707973131463622273
