| L(s) = 1 | + 3-s − 2·9-s + 7·11-s − 5·13-s + 17-s − 4·19-s − 6·23-s + 3·27-s + 3·29-s − 10·31-s + 7·33-s + 18·37-s − 5·39-s − 18·41-s + 9·47-s + 51-s + 10·53-s − 4·57-s − 6·59-s − 24·61-s − 10·67-s − 6·69-s + 4·71-s + 6·73-s + 17·79-s + 8·81-s + 12·83-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 2/3·9-s + 2.11·11-s − 1.38·13-s + 0.242·17-s − 0.917·19-s − 1.25·23-s + 0.577·27-s + 0.557·29-s − 1.79·31-s + 1.21·33-s + 2.95·37-s − 0.800·39-s − 2.81·41-s + 1.31·47-s + 0.140·51-s + 1.37·53-s − 0.529·57-s − 0.781·59-s − 3.07·61-s − 1.22·67-s − 0.722·69-s + 0.474·71-s + 0.702·73-s + 1.91·79-s + 8/9·81-s + 1.31·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{9} \cdot 5^{6} \cdot 7^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{9} \cdot 5^{6} \cdot 7^{6}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.478691357\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.478691357\) |
| \(L(\frac{3}{2})\) |
|
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 | $S_4\times C_2$ | \( 1 - T + p T^{2} - 8 T^{3} + p^{2} T^{4} - p^{2} T^{5} + p^{3} T^{6} \) |
| 11 | $S_4\times C_2$ | \( 1 - 7 T + 41 T^{2} - 146 T^{3} + 41 p T^{4} - 7 p^{2} T^{5} + p^{3} T^{6} \) |
| 13 | $S_4\times C_2$ | \( 1 + 5 T + 17 T^{2} + 24 T^{3} + 17 p T^{4} + 5 p^{2} T^{5} + p^{3} T^{6} \) |
| 17 | $S_4\times C_2$ | \( 1 - T + 27 T^{2} - 54 T^{3} + 27 p T^{4} - p^{2} T^{5} + p^{3} T^{6} \) |
| 19 | $S_4\times C_2$ | \( 1 + 4 T + 43 T^{2} + 160 T^{3} + 43 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} \) |
| 23 | $S_4\times C_2$ | \( 1 + 6 T + 41 T^{2} + 140 T^{3} + 41 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 29 | $S_4\times C_2$ | \( 1 - 3 T + 15 T^{2} - 66 T^{3} + 15 p T^{4} - 3 p^{2} T^{5} + p^{3} T^{6} \) |
| 31 | $S_4\times C_2$ | \( 1 + 10 T + 101 T^{2} + 540 T^{3} + 101 p T^{4} + 10 p^{2} T^{5} + p^{3} T^{6} \) |
| 37 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{3} \) |
| 41 | $S_4\times C_2$ | \( 1 + 18 T + 191 T^{2} + 1388 T^{3} + 191 p T^{4} + 18 p^{2} T^{5} + p^{3} T^{6} \) |
| 43 | $S_4\times C_2$ | \( 1 + 89 T^{2} + 64 T^{3} + 89 p T^{4} + p^{3} T^{6} \) |
| 47 | $S_4\times C_2$ | \( 1 - 9 T + 93 T^{2} - 614 T^{3} + 93 p T^{4} - 9 p^{2} T^{5} + p^{3} T^{6} \) |
| 53 | $S_4\times C_2$ | \( 1 - 10 T + 115 T^{2} - 588 T^{3} + 115 p T^{4} - 10 p^{2} T^{5} + p^{3} T^{6} \) |
| 59 | $S_4\times C_2$ | \( 1 + 6 T + 99 T^{2} + 752 T^{3} + 99 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 61 | $S_4\times C_2$ | \( 1 + 24 T + 365 T^{2} + 3368 T^{3} + 365 p T^{4} + 24 p^{2} T^{5} + p^{3} T^{6} \) |
| 67 | $S_4\times C_2$ | \( 1 + 10 T + 137 T^{2} + 828 T^{3} + 137 p T^{4} + 10 p^{2} T^{5} + p^{3} T^{6} \) |
| 71 | $S_4\times C_2$ | \( 1 - 4 T + 193 T^{2} - 504 T^{3} + 193 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} \) |
| 73 | $S_4\times C_2$ | \( 1 - 6 T + 191 T^{2} - 740 T^{3} + 191 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 79 | $S_4\times C_2$ | \( 1 - 17 T + 205 T^{2} - 2138 T^{3} + 205 p T^{4} - 17 p^{2} T^{5} + p^{3} T^{6} \) |
| 83 | $S_4\times C_2$ | \( 1 - 12 T + 107 T^{2} - 1168 T^{3} + 107 p T^{4} - 12 p^{2} T^{5} + p^{3} T^{6} \) |
| 89 | $S_4\times C_2$ | \( 1 + 95 T^{2} + 464 T^{3} + 95 p T^{4} + p^{3} T^{6} \) |
| 97 | $S_4\times C_2$ | \( 1 - 9 T + 275 T^{2} - 1702 T^{3} + 275 p T^{4} - 9 p^{2} T^{5} + p^{3} T^{6} \) |
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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
−6.79105290325709672229702434486, −6.53607680233365156366883110254, −6.28232435994641961659356757374, −6.23921980443125795736078598291, −5.85614006335336605792913412113, −5.64799979077394002117628830921, −5.58499761646370068606585507215, −4.96566413246495286670841742572, −4.84452767398691984449136115788, −4.69068919735809779125547410646, −4.42466072854223204629616823265, −4.21004330862728433159948885309, −3.92075886820817359124220100642, −3.49589393895620774420691280951, −3.42362646941160246528002363843, −3.41456027361489309763597082002, −2.71711501961668166795343912599, −2.58664032109832487218357896167, −2.44342651877737499717707341946, −1.98545399180533493112973181813, −1.76897251139115482951720884556, −1.54986673847021639396573650215, −0.987634127608682630521915097866, −0.75552357921717616885710799060, −0.17850612965095899080159418512,
0.17850612965095899080159418512, 0.75552357921717616885710799060, 0.987634127608682630521915097866, 1.54986673847021639396573650215, 1.76897251139115482951720884556, 1.98545399180533493112973181813, 2.44342651877737499717707341946, 2.58664032109832487218357896167, 2.71711501961668166795343912599, 3.41456027361489309763597082002, 3.42362646941160246528002363843, 3.49589393895620774420691280951, 3.92075886820817359124220100642, 4.21004330862728433159948885309, 4.42466072854223204629616823265, 4.69068919735809779125547410646, 4.84452767398691984449136115788, 4.96566413246495286670841742572, 5.58499761646370068606585507215, 5.64799979077394002117628830921, 5.85614006335336605792913412113, 6.23921980443125795736078598291, 6.28232435994641961659356757374, 6.53607680233365156366883110254, 6.79105290325709672229702434486
