| L(s) = 1 | + 3-s − 2·5-s − 3·7-s − 9-s + 3·11-s − 8·13-s − 2·15-s + 9·17-s + 3·19-s − 3·21-s + 4·23-s − 2·25-s − 5·27-s − 11·29-s − 5·31-s + 3·33-s + 6·35-s − 4·37-s − 8·39-s + 11·41-s + 4·43-s + 2·45-s − 2·47-s + 6·49-s + 9·51-s − 9·53-s − 6·55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.894·5-s − 1.13·7-s − 1/3·9-s + 0.904·11-s − 2.21·13-s − 0.516·15-s + 2.18·17-s + 0.688·19-s − 0.654·21-s + 0.834·23-s − 2/5·25-s − 0.962·27-s − 2.04·29-s − 0.898·31-s + 0.522·33-s + 1.01·35-s − 0.657·37-s − 1.28·39-s + 1.71·41-s + 0.609·43-s + 0.298·45-s − 0.291·47-s + 6/7·49-s + 1.26·51-s − 1.23·53-s − 0.809·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{18} \cdot 7^{3} \cdot 19^{3}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{18} \cdot 7^{3} \cdot 19^{3}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.340653938\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.340653938\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 7 | $C_1$ | \( ( 1 + T )^{3} \) |
| 19 | $C_1$ | \( ( 1 - T )^{3} \) |
| good | 3 | $S_4\times C_2$ | \( 1 - T + 2 T^{2} + 2 T^{3} + 2 p T^{4} - p^{2} T^{5} + p^{3} T^{6} \) |
| 5 | $S_4\times C_2$ | \( 1 + 2 T + 6 T^{2} + 6 T^{3} + 6 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 11 | $S_4\times C_2$ | \( 1 - 3 T + 26 T^{2} - 46 T^{3} + 26 p T^{4} - 3 p^{2} T^{5} + p^{3} T^{6} \) |
| 13 | $S_4\times C_2$ | \( 1 + 8 T + 50 T^{2} + 192 T^{3} + 50 p T^{4} + 8 p^{2} T^{5} + p^{3} T^{6} \) |
| 17 | $S_4\times C_2$ | \( 1 - 9 T + 4 p T^{2} - 292 T^{3} + 4 p^{2} T^{4} - 9 p^{2} T^{5} + p^{3} T^{6} \) |
| 23 | $S_4\times C_2$ | \( 1 - 4 T + 64 T^{2} - 180 T^{3} + 64 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} \) |
| 29 | $S_4\times C_2$ | \( 1 + 11 T + 102 T^{2} + 636 T^{3} + 102 p T^{4} + 11 p^{2} T^{5} + p^{3} T^{6} \) |
| 31 | $S_4\times C_2$ | \( 1 + 5 T + 76 T^{2} + 314 T^{3} + 76 p T^{4} + 5 p^{2} T^{5} + p^{3} T^{6} \) |
| 37 | $S_4\times C_2$ | \( 1 + 4 T + 92 T^{2} + 246 T^{3} + 92 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} \) |
| 41 | $S_4\times C_2$ | \( 1 - 11 T + 110 T^{2} - 622 T^{3} + 110 p T^{4} - 11 p^{2} T^{5} + p^{3} T^{6} \) |
| 43 | $S_4\times C_2$ | \( 1 - 4 T + 37 T^{2} - 376 T^{3} + 37 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} \) |
| 47 | $S_4\times C_2$ | \( 1 + 2 T + 66 T^{2} + 372 T^{3} + 66 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 53 | $S_4\times C_2$ | \( 1 + 9 T + 120 T^{2} + 632 T^{3} + 120 p T^{4} + 9 p^{2} T^{5} + p^{3} T^{6} \) |
| 59 | $S_4\times C_2$ | \( 1 - 12 T + 132 T^{2} - 1308 T^{3} + 132 p T^{4} - 12 p^{2} T^{5} + p^{3} T^{6} \) |
| 61 | $S_4\times C_2$ | \( 1 - 2 T + 26 T^{2} - 42 T^{3} + 26 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 67 | $S_4\times C_2$ | \( 1 - 9 T + 218 T^{2} - 1192 T^{3} + 218 p T^{4} - 9 p^{2} T^{5} + p^{3} T^{6} \) |
| 71 | $S_4\times C_2$ | \( 1 + 158 T^{2} - 58 T^{3} + 158 p T^{4} + p^{3} T^{6} \) |
| 73 | $S_4\times C_2$ | \( 1 + 21 T + 302 T^{2} + 2764 T^{3} + 302 p T^{4} + 21 p^{2} T^{5} + p^{3} T^{6} \) |
| 79 | $S_4\times C_2$ | \( 1 - 6 T + 89 T^{2} + 68 T^{3} + 89 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 83 | $S_4\times C_2$ | \( 1 - 7 T + 204 T^{2} - 1062 T^{3} + 204 p T^{4} - 7 p^{2} T^{5} + p^{3} T^{6} \) |
| 89 | $S_4\times C_2$ | \( 1 + 18 T + 335 T^{2} + 3268 T^{3} + 335 p T^{4} + 18 p^{2} T^{5} + p^{3} T^{6} \) |
| 97 | $S_4\times C_2$ | \( 1 - 28 T + 330 T^{2} - 2896 T^{3} + 330 p T^{4} - 28 p^{2} T^{5} + p^{3} T^{6} \) |
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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
−7.19181693558475457057465620795, −6.61487667750844684262870826751, −6.46697120505320068380294690365, −6.21304288510182901975075719130, −5.83736391754173069462216679333, −5.66612198719148753506840354935, −5.60194634240205233720156755742, −5.17926863514407035360079750163, −5.07091069185580242922509838651, −4.85349783164951712195728069608, −4.35055237876135894059879165030, −4.20890690505100790948186631897, −3.72319369770100674091808217261, −3.71894719284193403835209244211, −3.62009514222440069080929156965, −3.24900645801641252767418413092, −2.95417738949078819218298199656, −2.71557733646255377679495739498, −2.54311832246423664398305855522, −1.97293460850486025101056358784, −1.96536593976643843346974860924, −1.34979862011024683108000362082, −1.10395185667699594294475763082, −0.52458235259035336005619217026, −0.24985435976138412993749454907,
0.24985435976138412993749454907, 0.52458235259035336005619217026, 1.10395185667699594294475763082, 1.34979862011024683108000362082, 1.96536593976643843346974860924, 1.97293460850486025101056358784, 2.54311832246423664398305855522, 2.71557733646255377679495739498, 2.95417738949078819218298199656, 3.24900645801641252767418413092, 3.62009514222440069080929156965, 3.71894719284193403835209244211, 3.72319369770100674091808217261, 4.20890690505100790948186631897, 4.35055237876135894059879165030, 4.85349783164951712195728069608, 5.07091069185580242922509838651, 5.17926863514407035360079750163, 5.60194634240205233720156755742, 5.66612198719148753506840354935, 5.83736391754173069462216679333, 6.21304288510182901975075719130, 6.46697120505320068380294690365, 6.61487667750844684262870826751, 7.19181693558475457057465620795
