| 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
| \(L(1)\) |
\(\approx\) |
\(10.71566916\) |
| \(L(\frac12)\) |
\(\approx\) |
\(10.71566916\) |
| \(L(\frac{3}{2})\) |
|
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
−7.32978423627467424013688552157, −7.11815814209827606200225796226, −6.72963229750414867187342172892, −6.54721214089077695127035378687, −6.25268404930591717321120551759, −6.24501352288689693323548966527, −5.94401848486141309948851984287, −5.46667630292574187337794733024, −5.37219531680335811788711992720, −5.26215391481375022956417017972, −4.94311231927924970412377331296, −4.31894768033417450120747218202, −4.05083959780421508074003029691, −3.90262342166238141138992939140, −3.76206294749080176934552479733, −3.72925792048554045041174889215, −3.08012350977213722019998616421, −2.94133150827190903071046108002, −2.73445458802144659098519688293, −2.09414714391385306480416515173, −1.98925222462742627933710565658, −1.45735418037543622038551920924, −1.22199050865960453352279794054, −0.75680612626105744959477088505, −0.70408705342062346366785951287,
0.70408705342062346366785951287, 0.75680612626105744959477088505, 1.22199050865960453352279794054, 1.45735418037543622038551920924, 1.98925222462742627933710565658, 2.09414714391385306480416515173, 2.73445458802144659098519688293, 2.94133150827190903071046108002, 3.08012350977213722019998616421, 3.72925792048554045041174889215, 3.76206294749080176934552479733, 3.90262342166238141138992939140, 4.05083959780421508074003029691, 4.31894768033417450120747218202, 4.94311231927924970412377331296, 5.26215391481375022956417017972, 5.37219531680335811788711992720, 5.46667630292574187337794733024, 5.94401848486141309948851984287, 6.24501352288689693323548966527, 6.25268404930591717321120551759, 6.54721214089077695127035378687, 6.72963229750414867187342172892, 7.11815814209827606200225796226, 7.32978423627467424013688552157
