| L(s) = 1 | − 2-s + 3·3-s − 3·4-s + 3·5-s − 3·6-s − 3·7-s + 4·8-s + 6·9-s − 3·10-s + 11-s − 9·12-s + 3·14-s + 9·15-s + 3·16-s − 7·17-s − 6·18-s + 3·19-s − 9·20-s − 9·21-s − 22-s − 12·23-s + 12·24-s − 9·25-s + 10·27-s + 9·28-s − 29-s − 9·30-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1.73·3-s − 3/2·4-s + 1.34·5-s − 1.22·6-s − 1.13·7-s + 1.41·8-s + 2·9-s − 0.948·10-s + 0.301·11-s − 2.59·12-s + 0.801·14-s + 2.32·15-s + 3/4·16-s − 1.69·17-s − 1.41·18-s + 0.688·19-s − 2.01·20-s − 1.96·21-s − 0.213·22-s − 2.50·23-s + 2.44·24-s − 9/5·25-s + 1.92·27-s + 1.70·28-s − 0.185·29-s − 1.64·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{3} \cdot 7^{3} \cdot 13^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & -\,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{3} \cdot 7^{3} \cdot 13^{6}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr=\mathstrut & -\,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 - T )^{3} \) |
| 7 | $C_1$ | \( ( 1 + T )^{3} \) |
| 13 | | \( 1 \) |
| good | 2 | $A_4\times C_2$ | \( 1 + T + p^{2} T^{2} + 3 T^{3} + p^{3} T^{4} + p^{2} T^{5} + p^{3} T^{6} \) |
| 5 | $C_2$ | \( ( 1 - T + p T^{2} )^{3} \) |
| 11 | $A_4\times C_2$ | \( 1 - T + 31 T^{2} - 21 T^{3} + 31 p T^{4} - p^{2} T^{5} + p^{3} T^{6} \) |
| 17 | $A_4\times C_2$ | \( 1 + 7 T + 65 T^{2} + 245 T^{3} + 65 p T^{4} + 7 p^{2} T^{5} + p^{3} T^{6} \) |
| 19 | $A_4\times C_2$ | \( 1 - 3 T + 39 T^{2} - 101 T^{3} + 39 p T^{4} - 3 p^{2} T^{5} + p^{3} T^{6} \) |
| 23 | $A_4\times C_2$ | \( 1 + 12 T + 110 T^{2} + 595 T^{3} + 110 p T^{4} + 12 p^{2} T^{5} + p^{3} T^{6} \) |
| 29 | $A_4\times C_2$ | \( 1 + T + 71 T^{2} + p T^{3} + 71 p T^{4} + p^{2} T^{5} + p^{3} T^{6} \) |
| 31 | $A_4\times C_2$ | \( 1 + T + 91 T^{2} + 61 T^{3} + 91 p T^{4} + p^{2} T^{5} + p^{3} T^{6} \) |
| 37 | $A_4\times C_2$ | \( 1 - 7 T + 90 T^{2} - 427 T^{3} + 90 p T^{4} - 7 p^{2} T^{5} + p^{3} T^{6} \) |
| 41 | $A_4\times C_2$ | \( 1 + 8 T + 100 T^{2} + 669 T^{3} + 100 p T^{4} + 8 p^{2} T^{5} + p^{3} T^{6} \) |
| 43 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{3} \) |
| 47 | $A_4\times C_2$ | \( 1 + 12 T + 126 T^{2} + 751 T^{3} + 126 p T^{4} + 12 p^{2} T^{5} + p^{3} T^{6} \) |
| 53 | $A_4\times C_2$ | \( 1 - T + 38 T^{2} + 343 T^{3} + 38 p T^{4} - p^{2} T^{5} + p^{3} T^{6} \) |
| 59 | $A_4\times C_2$ | \( 1 + 21 T + 296 T^{2} + 2569 T^{3} + 296 p T^{4} + 21 p^{2} T^{5} + p^{3} T^{6} \) |
| 61 | $A_4\times C_2$ | \( 1 + 20 T + 300 T^{2} + 2609 T^{3} + 300 p T^{4} + 20 p^{2} T^{5} + p^{3} T^{6} \) |
| 67 | $A_4\times C_2$ | \( 1 + 12 T + 228 T^{2} + 1581 T^{3} + 228 p T^{4} + 12 p^{2} T^{5} + p^{3} T^{6} \) |
| 71 | $A_4\times C_2$ | \( 1 - 6 T + 204 T^{2} - 825 T^{3} + 204 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 73 | $A_4\times C_2$ | \( 1 - 3 T + 201 T^{2} - 425 T^{3} + 201 p T^{4} - 3 p^{2} T^{5} + p^{3} T^{6} \) |
| 79 | $A_4\times C_2$ | \( 1 + 9 T + 173 T^{2} + 1463 T^{3} + 173 p T^{4} + 9 p^{2} T^{5} + p^{3} T^{6} \) |
| 83 | $A_4\times C_2$ | \( 1 - 2 T + 150 T^{2} - 345 T^{3} + 150 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 89 | $A_4\times C_2$ | \( 1 - 19 T + 322 T^{2} - 3411 T^{3} + 322 p T^{4} - 19 p^{2} T^{5} + p^{3} T^{6} \) |
| 97 | $A_4\times C_2$ | \( 1 - 16 T + 290 T^{2} - 2487 T^{3} + 290 p T^{4} - 16 p^{2} T^{5} + p^{3} T^{6} \) |
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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
−8.141781988870050321398202704280, −7.84128277608918740285988226790, −7.67092783960977655051848184201, −7.42946097036267108987855990927, −6.98432625091011843581074393297, −6.69101553755611473008014596496, −6.40660959414456474379640846537, −6.21717477134880380724555447469, −6.02166170630598113612975202848, −5.98849507566142622955790252806, −5.23389153704638988405430910067, −5.06801702151247049279639966073, −4.95304177481505241477311782088, −4.40608168156305638869263856846, −4.23139068460139460184824727998, −4.16630315273834277617163040257, −3.57293240397339221139627463013, −3.45382009472197406615556902288, −3.34595627729276658144384397504, −2.80252735507408116092758042448, −2.44918001509032608301565617999, −2.12177145077171970843276191175, −1.79242354433085439269528882812, −1.46206077749141236781653775166, −1.45367781556815534727264964997, 0, 0, 0,
1.45367781556815534727264964997, 1.46206077749141236781653775166, 1.79242354433085439269528882812, 2.12177145077171970843276191175, 2.44918001509032608301565617999, 2.80252735507408116092758042448, 3.34595627729276658144384397504, 3.45382009472197406615556902288, 3.57293240397339221139627463013, 4.16630315273834277617163040257, 4.23139068460139460184824727998, 4.40608168156305638869263856846, 4.95304177481505241477311782088, 5.06801702151247049279639966073, 5.23389153704638988405430910067, 5.98849507566142622955790252806, 6.02166170630598113612975202848, 6.21717477134880380724555447469, 6.40660959414456474379640846537, 6.69101553755611473008014596496, 6.98432625091011843581074393297, 7.42946097036267108987855990927, 7.67092783960977655051848184201, 7.84128277608918740285988226790, 8.141781988870050321398202704280
