| L(s) = 1 | + 3·3-s + 3·5-s + 6·9-s + 6·11-s + 4·13-s + 9·15-s + 12·17-s + 3·19-s − 4·23-s + 6·25-s + 10·27-s − 6·31-s + 18·33-s − 4·37-s + 12·39-s + 2·41-s + 2·43-s + 18·45-s + 4·47-s − 7·49-s + 36·51-s − 10·53-s + 18·55-s + 9·57-s + 2·59-s + 14·61-s + 12·65-s + ⋯ |
| L(s) = 1 | + 1.73·3-s + 1.34·5-s + 2·9-s + 1.80·11-s + 1.10·13-s + 2.32·15-s + 2.91·17-s + 0.688·19-s − 0.834·23-s + 6/5·25-s + 1.92·27-s − 1.07·31-s + 3.13·33-s − 0.657·37-s + 1.92·39-s + 0.312·41-s + 0.304·43-s + 2.68·45-s + 0.583·47-s − 49-s + 5.04·51-s − 1.37·53-s + 2.42·55-s + 1.19·57-s + 0.260·59-s + 1.79·61-s + 1.48·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{9} \cdot 3^{3} \cdot 5^{3} \cdot 19^{3}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{9} \cdot 3^{3} \cdot 5^{3} \cdot 19^{3}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(18.29781533\) |
| \(L(\frac12)\) |
\(\approx\) |
\(18.29781533\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 3 | $C_1$ | \( ( 1 - T )^{3} \) |
| 5 | $C_1$ | \( ( 1 - T )^{3} \) |
| 19 | $C_1$ | \( ( 1 - T )^{3} \) |
| good | 7 | $S_4\times C_2$ | \( 1 + p T^{2} + 12 T^{3} + p^{2} T^{4} + p^{3} T^{6} \) |
| 11 | $S_4\times C_2$ | \( 1 - 6 T + 31 T^{2} - 100 T^{3} + 31 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 13 | $S_4\times C_2$ | \( 1 - 4 T + T^{2} + 60 T^{3} + p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} \) |
| 17 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{3} \) |
| 23 | $S_4\times C_2$ | \( 1 + 4 T + 25 T^{2} + 152 T^{3} + 25 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} \) |
| 29 | $S_4\times C_2$ | \( 1 + 73 T^{2} - 12 T^{3} + 73 p T^{4} + p^{3} T^{6} \) |
| 31 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{3} \) |
| 37 | $S_4\times C_2$ | \( 1 + 4 T + 33 T^{2} + 476 T^{3} + 33 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} \) |
| 41 | $S_4\times C_2$ | \( 1 - 2 T + p T^{2} + 180 T^{3} + p^{2} T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 43 | $S_4\times C_2$ | \( 1 - 2 T + 87 T^{2} - 252 T^{3} + 87 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 47 | $S_4\times C_2$ | \( 1 - 4 T + 97 T^{2} - 344 T^{3} + 97 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} \) |
| 53 | $S_4\times C_2$ | \( 1 + 10 T + 143 T^{2} + 964 T^{3} + 143 p T^{4} + 10 p^{2} T^{5} + p^{3} T^{6} \) |
| 59 | $S_4\times C_2$ | \( 1 - 2 T + 33 T^{2} + 452 T^{3} + 33 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 61 | $S_4\times C_2$ | \( 1 - 14 T + 3 p T^{2} - 1564 T^{3} + 3 p^{2} T^{4} - 14 p^{2} T^{5} + p^{3} T^{6} \) |
| 67 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{3} \) |
| 71 | $S_4\times C_2$ | \( 1 + 4 T + 45 T^{2} + 1208 T^{3} + 45 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} \) |
| 73 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{3} \) |
| 79 | $S_4\times C_2$ | \( 1 + 2 T + 45 T^{2} - 708 T^{3} + 45 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 83 | $S_4\times C_2$ | \( 1 + 6 T + 205 T^{2} + 796 T^{3} + 205 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 89 | $S_4\times C_2$ | \( 1 - 10 T + 217 T^{2} - 1828 T^{3} + 217 p T^{4} - 10 p^{2} T^{5} + p^{3} T^{6} \) |
| 97 | $S_4\times C_2$ | \( 1 - 32 T + 549 T^{2} - 6244 T^{3} + 549 p T^{4} - 32 p^{2} T^{5} + p^{3} T^{6} \) |
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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
−8.239949368048067660928029858659, −7.65785342906162182691483641053, −7.54753381518520507436565797650, −7.42147347651069445689709996004, −6.93758529326072797897469341671, −6.64845931668855687924510046732, −6.54465666208020730017522988391, −6.06156714952640329213099948362, −5.92843587225867296271803274547, −5.79649303451895256750849248245, −5.15421895437659861579351049341, −5.04697021534655749578656915407, −5.01700259282724262910992683411, −4.17809954177998150113197862138, −3.91463183154030717323806723555, −3.81111709892467254342720053276, −3.47446918324216801461481856338, −3.25983494386567445117791461734, −3.07891518670211636670922578701, −2.37464194926560177328707096247, −2.25359595836656154338449864706, −1.77513826055931309509211154237, −1.45135511230001580958708003342, −1.09868585059946167815270153764, −0.932664508981184346157939558182,
0.932664508981184346157939558182, 1.09868585059946167815270153764, 1.45135511230001580958708003342, 1.77513826055931309509211154237, 2.25359595836656154338449864706, 2.37464194926560177328707096247, 3.07891518670211636670922578701, 3.25983494386567445117791461734, 3.47446918324216801461481856338, 3.81111709892467254342720053276, 3.91463183154030717323806723555, 4.17809954177998150113197862138, 5.01700259282724262910992683411, 5.04697021534655749578656915407, 5.15421895437659861579351049341, 5.79649303451895256750849248245, 5.92843587225867296271803274547, 6.06156714952640329213099948362, 6.54465666208020730017522988391, 6.64845931668855687924510046732, 6.93758529326072797897469341671, 7.42147347651069445689709996004, 7.54753381518520507436565797650, 7.65785342906162182691483641053, 8.239949368048067660928029858659
