| L(s) = 1 | + 2-s + 3·3-s − 4-s + 3·5-s + 3·6-s − 3·7-s − 8-s + 6·9-s + 3·10-s + 3·11-s − 3·12-s + 8·13-s − 3·14-s + 9·15-s − 16-s + 8·17-s + 6·18-s − 2·19-s − 3·20-s − 9·21-s + 3·22-s − 6·23-s − 3·24-s + 6·25-s + 8·26-s + 10·27-s + 3·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1.73·3-s − 1/2·4-s + 1.34·5-s + 1.22·6-s − 1.13·7-s − 0.353·8-s + 2·9-s + 0.948·10-s + 0.904·11-s − 0.866·12-s + 2.21·13-s − 0.801·14-s + 2.32·15-s − 1/4·16-s + 1.94·17-s + 1.41·18-s − 0.458·19-s − 0.670·20-s − 1.96·21-s + 0.639·22-s − 1.25·23-s − 0.612·24-s + 6/5·25-s + 1.56·26-s + 1.92·27-s + 0.566·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{3} \cdot 5^{3} \cdot 7^{3} \cdot 11^{3}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{3} \cdot 5^{3} \cdot 7^{3} \cdot 11^{3}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(12.53864897\) |
| \(L(\frac12)\) |
\(\approx\) |
\(12.53864897\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 - T )^{3} \) |
| 5 | $C_1$ | \( ( 1 - T )^{3} \) |
| 7 | $C_1$ | \( ( 1 + T )^{3} \) |
| 11 | $C_1$ | \( ( 1 - T )^{3} \) |
| good | 2 | $S_4\times C_2$ | \( 1 - T + p T^{2} - p T^{3} + p^{2} T^{4} - p^{2} T^{5} + p^{3} T^{6} \) |
| 13 | $S_4\times C_2$ | \( 1 - 8 T + 53 T^{2} - 204 T^{3} + 53 p T^{4} - 8 p^{2} T^{5} + p^{3} T^{6} \) |
| 17 | $S_4\times C_2$ | \( 1 - 8 T + 56 T^{2} - 226 T^{3} + 56 p T^{4} - 8 p^{2} T^{5} + p^{3} T^{6} \) |
| 19 | $S_4\times C_2$ | \( 1 + 2 T + 26 T^{2} + 120 T^{3} + 26 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 23 | $S_4\times C_2$ | \( 1 + 6 T + 74 T^{2} + 268 T^{3} + 74 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 29 | $S_4\times C_2$ | \( 1 - 10 T + 4 p T^{2} - 602 T^{3} + 4 p^{2} T^{4} - 10 p^{2} T^{5} + p^{3} T^{6} \) |
| 31 | $S_4\times C_2$ | \( 1 + 2 T + 79 T^{2} + 92 T^{3} + 79 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 37 | $S_4\times C_2$ | \( 1 - 8 T + 53 T^{2} - 300 T^{3} + 53 p T^{4} - 8 p^{2} T^{5} + p^{3} T^{6} \) |
| 41 | $S_4\times C_2$ | \( 1 - 4 T + 73 T^{2} - 396 T^{3} + 73 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} \) |
| 43 | $S_4\times C_2$ | \( 1 + 2 p T^{2} - 44 T^{3} + 2 p^{2} T^{4} + p^{3} T^{6} \) |
| 47 | $S_4\times C_2$ | \( 1 - 2 T + 127 T^{2} - 156 T^{3} + 127 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 53 | $S_4\times C_2$ | \( 1 + 152 T^{2} + 2 T^{3} + 152 p T^{4} + p^{3} T^{6} \) |
| 59 | $S_4\times C_2$ | \( 1 + 12 T + 186 T^{2} + 1340 T^{3} + 186 p T^{4} + 12 p^{2} T^{5} + p^{3} T^{6} \) |
| 61 | $S_4\times C_2$ | \( 1 - 4 T + 132 T^{2} - 262 T^{3} + 132 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} \) |
| 67 | $S_4\times C_2$ | \( 1 + 173 T^{2} - 16 T^{3} + 173 p T^{4} + p^{3} T^{6} \) |
| 71 | $S_4\times C_2$ | \( 1 + 18 T + 255 T^{2} + 2284 T^{3} + 255 p T^{4} + 18 p^{2} T^{5} + p^{3} T^{6} \) |
| 73 | $S_4\times C_2$ | \( 1 - 14 T + 267 T^{2} - 2060 T^{3} + 267 p T^{4} - 14 p^{2} T^{5} + p^{3} T^{6} \) |
| 79 | $S_4\times C_2$ | \( 1 - 2 T - 9 T^{2} + 292 T^{3} - 9 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 83 | $S_4\times C_2$ | \( 1 - 6 T + 114 T^{2} - 1288 T^{3} + 114 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 89 | $S_4\times C_2$ | \( 1 + 10 T + 200 T^{2} + 1778 T^{3} + 200 p T^{4} + 10 p^{2} T^{5} + p^{3} T^{6} \) |
| 97 | $S_4\times C_2$ | \( 1 - 10 T + 84 T^{2} - 118 T^{3} + 84 p T^{4} - 10 p^{2} T^{5} + p^{3} T^{6} \) |
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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
−8.856721954676514406330169936037, −8.349445127491029847310309488632, −8.205201349979257110692197035165, −8.144936645314817696902072406783, −7.60713544285575183569769036019, −7.46908858325147721820875227971, −6.82700770885754097021968864876, −6.69302223569455262545363937945, −6.31666492078190340350377761070, −6.25876079029069407740083361424, −5.85145447111588207445494802582, −5.51703933870808662882755483247, −5.51321268685974033765276187276, −4.53132558662130568780621810578, −4.45105020689498713526789056964, −4.42658302705657473779611412073, −3.69417634297439624050722858446, −3.58181207372389250448723130375, −3.40965620646151300683365652535, −2.84260305159584516820937983815, −2.74827704447577099792691429509, −2.10002421531779338431598503708, −1.74846656866396397667162548706, −1.08756302667123128444735616533, −1.00903709394830321841901430407,
1.00903709394830321841901430407, 1.08756302667123128444735616533, 1.74846656866396397667162548706, 2.10002421531779338431598503708, 2.74827704447577099792691429509, 2.84260305159584516820937983815, 3.40965620646151300683365652535, 3.58181207372389250448723130375, 3.69417634297439624050722858446, 4.42658302705657473779611412073, 4.45105020689498713526789056964, 4.53132558662130568780621810578, 5.51321268685974033765276187276, 5.51703933870808662882755483247, 5.85145447111588207445494802582, 6.25876079029069407740083361424, 6.31666492078190340350377761070, 6.69302223569455262545363937945, 6.82700770885754097021968864876, 7.46908858325147721820875227971, 7.60713544285575183569769036019, 8.144936645314817696902072406783, 8.205201349979257110692197035165, 8.349445127491029847310309488632, 8.856721954676514406330169936037
