| L(s) = 1 | − 3·3-s − 6·11-s + 6·17-s − 6·19-s + 3·23-s + 12·27-s + 12·29-s + 3·31-s + 18·33-s − 6·37-s − 3·41-s − 9·43-s − 18·47-s − 18·51-s + 6·53-s + 18·57-s + 3·59-s − 9·61-s + 12·67-s − 9·69-s + 9·71-s + 12·73-s − 3·79-s − 18·81-s − 36·87-s − 6·89-s − 9·93-s + ⋯ |
| L(s) = 1 | − 1.73·3-s − 1.80·11-s + 1.45·17-s − 1.37·19-s + 0.625·23-s + 2.30·27-s + 2.22·29-s + 0.538·31-s + 3.13·33-s − 0.986·37-s − 0.468·41-s − 1.37·43-s − 2.62·47-s − 2.52·51-s + 0.824·53-s + 2.38·57-s + 0.390·59-s − 1.15·61-s + 1.46·67-s − 1.08·69-s + 1.06·71-s + 1.40·73-s − 0.337·79-s − 2·81-s − 3.85·87-s − 0.635·89-s − 0.933·93-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{9} \cdot 5^{6} \cdot 7^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & -\,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{9} \cdot 5^{6} \cdot 7^{6}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr=\mathstrut & -\,\Lambda(1-s)\end{aligned}\]
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 | 2 | | \( 1 \) |
| 5 | | \( 1 \) |
| 7 | | \( 1 \) |
| good | 3 | $A_4\times C_2$ | \( 1 + p T + p^{2} T^{2} + 5 p T^{3} + p^{3} T^{4} + p^{3} T^{5} + p^{3} T^{6} \) |
| 11 | $A_4\times C_2$ | \( 1 + 6 T + 36 T^{2} + 113 T^{3} + 36 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 13 | $C_6$ | \( 1 - 19 T^{3} + p^{3} T^{6} \) |
| 17 | $A_4\times C_2$ | \( 1 - 6 T + 36 T^{2} - 185 T^{3} + 36 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 19 | $A_4\times C_2$ | \( 1 + 6 T + 48 T^{2} + 211 T^{3} + 48 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 23 | $A_4\times C_2$ | \( 1 - 3 T + 51 T^{2} - 155 T^{3} + 51 p T^{4} - 3 p^{2} T^{5} + p^{3} T^{6} \) |
| 29 | $A_4\times C_2$ | \( 1 - 12 T + 126 T^{2} - 715 T^{3} + 126 p T^{4} - 12 p^{2} T^{5} + p^{3} T^{6} \) |
| 31 | $A_4\times C_2$ | \( 1 - 3 T + 57 T^{2} - 167 T^{3} + 57 p T^{4} - 3 p^{2} T^{5} + p^{3} T^{6} \) |
| 37 | $A_4\times C_2$ | \( 1 + 6 T + 84 T^{2} + 393 T^{3} + 84 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 41 | $A_4\times C_2$ | \( 1 + 3 T + 105 T^{2} + 189 T^{3} + 105 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} \) |
| 43 | $A_4\times C_2$ | \( 1 + 9 T + 117 T^{2} + 595 T^{3} + 117 p T^{4} + 9 p^{2} T^{5} + p^{3} T^{6} \) |
| 47 | $A_4\times C_2$ | \( 1 + 18 T + 228 T^{2} + 1799 T^{3} + 228 p T^{4} + 18 p^{2} T^{5} + p^{3} T^{6} \) |
| 53 | $A_4\times C_2$ | \( 1 - 6 T - 12 T^{2} + 441 T^{3} - 12 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 59 | $A_4\times C_2$ | \( 1 - 3 T + 153 T^{2} - 301 T^{3} + 153 p T^{4} - 3 p^{2} T^{5} + p^{3} T^{6} \) |
| 61 | $A_4\times C_2$ | \( 1 + 9 T + 117 T^{2} + 1135 T^{3} + 117 p T^{4} + 9 p^{2} T^{5} + p^{3} T^{6} \) |
| 67 | $A_4\times C_2$ | \( 1 - 12 T + 141 T^{2} - 1024 T^{3} + 141 p T^{4} - 12 p^{2} T^{5} + p^{3} T^{6} \) |
| 71 | $A_4\times C_2$ | \( 1 - 9 T + 132 T^{2} - 765 T^{3} + 132 p T^{4} - 9 p^{2} T^{5} + p^{3} T^{6} \) |
| 73 | $A_4\times C_2$ | \( 1 - 12 T + 3 p T^{2} - 1560 T^{3} + 3 p^{2} T^{4} - 12 p^{2} T^{5} + p^{3} T^{6} \) |
| 79 | $A_4\times C_2$ | \( 1 + 3 T + 3 p T^{2} + 471 T^{3} + 3 p^{2} T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} \) |
| 83 | $A_4\times C_2$ | \( 1 + 192 T^{2} + 163 T^{3} + 192 p T^{4} + p^{3} T^{6} \) |
| 89 | $A_4\times C_2$ | \( 1 + 6 T + 150 T^{2} + 369 T^{3} + 150 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 97 | $A_4\times C_2$ | \( 1 - 21 T + 357 T^{2} - 3607 T^{3} + 357 p T^{4} - 21 p^{2} T^{5} + p^{3} T^{6} \) |
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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
−7.02952304452058724584864392355, −6.65309713205696532755117639053, −6.59036873619920650301050283255, −6.56838860280185303976787371920, −6.04591000636291945835791148226, −6.02807234855834406109835748668, −5.74856322302983811081506321166, −5.37412612683603453426028224073, −5.36989104848644578812466391579, −5.03025544903382334102129415736, −4.84786207239802080645521077026, −4.80722370111065010834329869004, −4.65354387789043841084178215480, −4.07933626771460536347746509601, −3.77556959055943832267274248422, −3.53093901855616667422384263793, −3.20383294196903007490161382417, −3.00096878073078204347950709144, −2.87138610498256404241843862721, −2.39738155117728233733932709533, −2.19752143422725866760710999568, −2.00220506698059731288563630351, −1.24807193574017426427664470653, −1.13131983885756149735412009720, −0.856911753956159264937477492611, 0, 0, 0,
0.856911753956159264937477492611, 1.13131983885756149735412009720, 1.24807193574017426427664470653, 2.00220506698059731288563630351, 2.19752143422725866760710999568, 2.39738155117728233733932709533, 2.87138610498256404241843862721, 3.00096878073078204347950709144, 3.20383294196903007490161382417, 3.53093901855616667422384263793, 3.77556959055943832267274248422, 4.07933626771460536347746509601, 4.65354387789043841084178215480, 4.80722370111065010834329869004, 4.84786207239802080645521077026, 5.03025544903382334102129415736, 5.36989104848644578812466391579, 5.37412612683603453426028224073, 5.74856322302983811081506321166, 6.02807234855834406109835748668, 6.04591000636291945835791148226, 6.56838860280185303976787371920, 6.59036873619920650301050283255, 6.65309713205696532755117639053, 7.02952304452058724584864392355
