| L(s) = 1 | − 3-s − 2·5-s − 3·7-s + 9-s + 7·11-s + 2·15-s + 17-s − 3·19-s + 3·21-s + 16·23-s − 2·25-s + 5·27-s + 29-s − 5·31-s − 7·33-s + 6·35-s + 6·37-s + 41-s − 4·43-s − 2·45-s − 14·47-s + 6·49-s − 51-s − 31·53-s − 14·55-s + 3·57-s + 18·59-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.894·5-s − 1.13·7-s + 1/3·9-s + 2.11·11-s + 0.516·15-s + 0.242·17-s − 0.688·19-s + 0.654·21-s + 3.33·23-s − 2/5·25-s + 0.962·27-s + 0.185·29-s − 0.898·31-s − 1.21·33-s + 1.01·35-s + 0.986·37-s + 0.156·41-s − 0.609·43-s − 0.298·45-s − 2.04·47-s + 6/7·49-s − 0.140·51-s − 4.25·53-s − 1.88·55-s + 0.397·57-s + 2.34·59-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
| \(L(1)\) |
\(\approx\) |
\(2.284145954\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.284145954\) |
| \(L(\frac{3}{2})\) |
|
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 | $S_4\times C_2$ | \( 1 + T - 2 p T^{3} + p^{2} T^{5} + p^{3} T^{6} \) |
| 5 | $S_4\times C_2$ | \( 1 + 2 T + 6 T^{2} + 14 T^{3} + 6 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 11 | $S_4\times C_2$ | \( 1 - 7 T + 34 T^{2} - 118 T^{3} + 34 p T^{4} - 7 p^{2} T^{5} + p^{3} T^{6} \) |
| 13 | $D_{6}$ | \( 1 - 54 T^{3} + p^{3} T^{6} \) |
| 17 | $S_4\times C_2$ | \( 1 - T + 36 T^{2} - 16 T^{3} + 36 p T^{4} - p^{2} T^{5} + p^{3} T^{6} \) |
| 23 | $S_4\times C_2$ | \( 1 - 16 T + 144 T^{2} - 832 T^{3} + 144 p T^{4} - 16 p^{2} T^{5} + p^{3} T^{6} \) |
| 29 | $S_4\times C_2$ | \( 1 - T + 72 T^{2} - 40 T^{3} + 72 p T^{4} - p^{2} T^{5} + p^{3} T^{6} \) |
| 31 | $S_4\times C_2$ | \( 1 + 5 T + 86 T^{2} + 302 T^{3} + 86 p T^{4} + 5 p^{2} T^{5} + p^{3} T^{6} \) |
| 37 | $S_4\times C_2$ | \( 1 - 6 T + 30 T^{2} - 282 T^{3} + 30 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 41 | $S_4\times C_2$ | \( 1 - T + 42 T^{2} + 200 T^{3} + 42 p T^{4} - p^{2} T^{5} + p^{3} T^{6} \) |
| 43 | $S_4\times C_2$ | \( 1 + 4 T + 93 T^{2} + 296 T^{3} + 93 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} \) |
| 47 | $S_4\times C_2$ | \( 1 + 14 T + 150 T^{2} + 1100 T^{3} + 150 p T^{4} + 14 p^{2} T^{5} + p^{3} T^{6} \) |
| 53 | $S_4\times C_2$ | \( 1 + 31 T + 470 T^{2} + 4284 T^{3} + 470 p T^{4} + 31 p^{2} T^{5} + p^{3} T^{6} \) |
| 59 | $S_4\times C_2$ | \( 1 - 18 T + 246 T^{2} - 2160 T^{3} + 246 p T^{4} - 18 p^{2} T^{5} + p^{3} T^{6} \) |
| 61 | $S_4\times C_2$ | \( 1 + 26 T + 398 T^{2} + 3734 T^{3} + 398 p T^{4} + 26 p^{2} T^{5} + p^{3} T^{6} \) |
| 67 | $S_4\times C_2$ | \( 1 + T + 94 T^{2} + 110 T^{3} + 94 p T^{4} + p^{2} T^{5} + p^{3} T^{6} \) |
| 71 | $S_4\times C_2$ | \( 1 - 12 T + 168 T^{2} - 1380 T^{3} + 168 p T^{4} - 12 p^{2} T^{5} + p^{3} T^{6} \) |
| 73 | $S_4\times C_2$ | \( 1 - 19 T + 330 T^{2} - 2960 T^{3} + 330 p T^{4} - 19 p^{2} T^{5} + p^{3} T^{6} \) |
| 79 | $S_4\times C_2$ | \( 1 + 4 T + 201 T^{2} + 584 T^{3} + 201 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} \) |
| 83 | $S_4\times C_2$ | \( 1 + 13 T + 4 T^{2} - 1070 T^{3} + 4 p T^{4} + 13 p^{2} T^{5} + p^{3} T^{6} \) |
| 89 | $S_4\times C_2$ | \( 1 + 12 T + 147 T^{2} + 704 T^{3} + 147 p T^{4} + 12 p^{2} T^{5} + p^{3} T^{6} \) |
| 97 | $S_4\times C_2$ | \( 1 + 8 T + 156 T^{2} + 1906 T^{3} + 156 p T^{4} + 8 p^{2} T^{5} + p^{3} T^{6} \) |
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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
−6.91642147420745210133188146587, −6.62184419204002041063340920303, −6.52336016123654829274922242743, −6.34052664001993887919436478307, −5.94184918207743366799289190590, −5.77540459115707175287224979171, −5.65637010065159159267056683458, −5.11382744227814027892088936851, −4.86180031944613324352861532458, −4.65117923035807252008061452378, −4.64520891481843036013958086628, −4.38123619603899161848870084130, −3.87311867729149724299568554032, −3.73640320085587360343548950637, −3.47452596270728101377169845472, −3.31485518232009150812517471510, −2.96045661514207416379855030556, −2.86258683528650375381443170712, −2.52825694833560734923918330967, −1.87072492040395193758747388314, −1.62713835083457126486192726010, −1.25016092033467334981864045882, −1.22135659841066034019150866158, −0.44431870555185554149840921430, −0.43498480492521483710492847745,
0.43498480492521483710492847745, 0.44431870555185554149840921430, 1.22135659841066034019150866158, 1.25016092033467334981864045882, 1.62713835083457126486192726010, 1.87072492040395193758747388314, 2.52825694833560734923918330967, 2.86258683528650375381443170712, 2.96045661514207416379855030556, 3.31485518232009150812517471510, 3.47452596270728101377169845472, 3.73640320085587360343548950637, 3.87311867729149724299568554032, 4.38123619603899161848870084130, 4.64520891481843036013958086628, 4.65117923035807252008061452378, 4.86180031944613324352861532458, 5.11382744227814027892088936851, 5.65637010065159159267056683458, 5.77540459115707175287224979171, 5.94184918207743366799289190590, 6.34052664001993887919436478307, 6.52336016123654829274922242743, 6.62184419204002041063340920303, 6.91642147420745210133188146587
