| L(s) = 1 | − 3·2-s + 6·4-s − 10·8-s + 15·16-s − 21·32-s + 3·41-s + 3·49-s + 3·53-s + 28·64-s − 9·82-s − 3·97-s − 9·98-s + 3·101-s − 9·106-s + 3·121-s + 125-s + ⋯ |
| L(s) = 1 | − 3·2-s + 6·4-s − 10·8-s + 15·16-s − 21·32-s + 3·41-s + 3·49-s + 3·53-s + 28·64-s − 9·82-s − 3·97-s − 9·98-s + 3·101-s − 9·106-s + 3·121-s + 125-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{6} \cdot 3^{18}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{6} \cdot 3^{18}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(0.4196857704\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4196857704\) |
| \(L(1)\) |
|
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 | $C_1$ | \( ( 1 + T )^{3} \) |
| 3 | | \( 1 \) |
| good | 5 | $C_6$ | \( 1 - T^{3} + T^{6} \) |
| 7 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{3}( 1 + T )^{3} \) |
| 11 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{3}( 1 + T )^{3} \) |
| 13 | $C_6$ | \( 1 + T^{3} + T^{6} \) |
| 17 | $C_6$ | \( 1 - T^{3} + T^{6} \) |
| 19 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{3}( 1 + T )^{3} \) |
| 23 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{3}( 1 + T )^{3} \) |
| 29 | $C_6$ | \( 1 - T^{3} + T^{6} \) |
| 31 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{3}( 1 + T )^{3} \) |
| 37 | $C_6$ | \( 1 + T^{3} + T^{6} \) |
| 41 | $C_2$ | \( ( 1 - T + T^{2} )^{3} \) |
| 43 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{3}( 1 + T )^{3} \) |
| 47 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{3}( 1 + T )^{3} \) |
| 53 | $C_2$ | \( ( 1 - T + T^{2} )^{3} \) |
| 59 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{3}( 1 + T )^{3} \) |
| 61 | $C_6$ | \( 1 + T^{3} + T^{6} \) |
| 67 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{3}( 1 + T )^{3} \) |
| 71 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{3}( 1 + T )^{3} \) |
| 73 | $C_6$ | \( 1 + T^{3} + T^{6} \) |
| 79 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{3}( 1 + T )^{3} \) |
| 83 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{3}( 1 + T )^{3} \) |
| 89 | $C_6$ | \( 1 - T^{3} + T^{6} \) |
| 97 | $C_2$ | \( ( 1 + T + T^{2} )^{3} \) |
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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
−8.170313978896700103047047775062, −7.60530292378119634323509758882, −7.45612170493658376192948824903, −7.39632321222861403033386589807, −7.13737715197232819549613671000, −6.93298805025348307662276439779, −6.65157014736788189479797875074, −6.18048347962079642011481794209, −6.04642032093696833327729983531, −5.86156311357571870920210838371, −5.50748082439024894203809636969, −5.47549087700424298706653488579, −4.95127056019875205315849057263, −4.25489296013916112985976635813, −4.22668190298031973547130304374, −3.80421022166479137950161263206, −3.41014288895231580201693849629, −3.08508958876946500393768504401, −2.73500645403033927651433416048, −2.33821405111217117216615166135, −2.21231087245752853892735231660, −2.03422997130692547851069777448, −1.21973577334865647129708500048, −1.00108169768533622710915083698, −0.67310729882402486125261143500,
0.67310729882402486125261143500, 1.00108169768533622710915083698, 1.21973577334865647129708500048, 2.03422997130692547851069777448, 2.21231087245752853892735231660, 2.33821405111217117216615166135, 2.73500645403033927651433416048, 3.08508958876946500393768504401, 3.41014288895231580201693849629, 3.80421022166479137950161263206, 4.22668190298031973547130304374, 4.25489296013916112985976635813, 4.95127056019875205315849057263, 5.47549087700424298706653488579, 5.50748082439024894203809636969, 5.86156311357571870920210838371, 6.04642032093696833327729983531, 6.18048347962079642011481794209, 6.65157014736788189479797875074, 6.93298805025348307662276439779, 7.13737715197232819549613671000, 7.39632321222861403033386589807, 7.45612170493658376192948824903, 7.60530292378119634323509758882, 8.170313978896700103047047775062
