| L(s) = 1 | + 3·3-s + 3·5-s + 2·7-s + 6·9-s + 4·11-s + 9·15-s + 6·17-s + 4·19-s + 6·21-s − 4·23-s + 6·25-s + 10·27-s + 6·29-s + 6·31-s + 12·33-s + 6·35-s − 4·37-s + 10·41-s + 4·43-s + 18·45-s − 4·47-s − 49-s + 18·51-s + 6·53-s + 12·55-s + 12·57-s + 8·59-s + ⋯ |
| L(s) = 1 | + 1.73·3-s + 1.34·5-s + 0.755·7-s + 2·9-s + 1.20·11-s + 2.32·15-s + 1.45·17-s + 0.917·19-s + 1.30·21-s − 0.834·23-s + 6/5·25-s + 1.92·27-s + 1.11·29-s + 1.07·31-s + 2.08·33-s + 1.01·35-s − 0.657·37-s + 1.56·41-s + 0.609·43-s + 2.68·45-s − 0.583·47-s − 1/7·49-s + 2.52·51-s + 0.824·53-s + 1.61·55-s + 1.58·57-s + 1.04·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{3} \cdot 5^{3}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{3} \cdot 5^{3}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(24.58287379\) |
| \(L(\frac12)\) |
\(\approx\) |
\(24.58287379\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 - T )^{3} \) |
| good | 7 | $S_4\times C_2$ | \( 1 - 2 T + 5 T^{2} - 12 T^{3} + 5 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 11 | $S_4\times C_2$ | \( 1 - 4 T + 9 T^{2} - 24 T^{3} + 9 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} \) |
| 13 | $S_4\times C_2$ | \( 1 + 11 T^{2} - 16 T^{3} + 11 p T^{4} + p^{3} T^{6} \) |
| 17 | $S_4\times C_2$ | \( 1 - 6 T + 35 T^{2} - 172 T^{3} + 35 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 19 | $S_4\times C_2$ | \( 1 - 4 T + 45 T^{2} - 136 T^{3} + 45 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} \) |
| 23 | $S_4\times C_2$ | \( 1 + 4 T + 57 T^{2} + 168 T^{3} + 57 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} \) |
| 29 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{3} \) |
| 31 | $S_4\times C_2$ | \( 1 - 6 T + 77 T^{2} - 308 T^{3} + 77 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 37 | $S_4\times C_2$ | \( 1 + 4 T + 51 T^{2} + 40 T^{3} + 51 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} \) |
| 41 | $S_4\times C_2$ | \( 1 - 10 T + 87 T^{2} - 588 T^{3} + 87 p T^{4} - 10 p^{2} T^{5} + p^{3} T^{6} \) |
| 43 | $S_4\times C_2$ | \( 1 - 4 T + 65 T^{2} - 216 T^{3} + 65 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} \) |
| 47 | $S_4\times C_2$ | \( 1 + 4 T + 49 T^{2} - 120 T^{3} + 49 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} \) |
| 53 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{3} \) |
| 59 | $S_4\times C_2$ | \( 1 - 8 T + 169 T^{2} - 912 T^{3} + 169 p T^{4} - 8 p^{2} T^{5} + p^{3} T^{6} \) |
| 61 | $S_4\times C_2$ | \( 1 - 8 T + 87 T^{2} - 464 T^{3} + 87 p T^{4} - 8 p^{2} T^{5} + p^{3} T^{6} \) |
| 67 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{3} \) |
| 71 | $S_4\times C_2$ | \( 1 + 4 T + 101 T^{2} + 632 T^{3} + 101 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} \) |
| 73 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{3} \) |
| 79 | $S_4\times C_2$ | \( 1 + 18 T + 317 T^{2} + 2908 T^{3} + 317 p T^{4} + 18 p^{2} T^{5} + p^{3} T^{6} \) |
| 83 | $S_4\times C_2$ | \( 1 - 16 T + 265 T^{2} - 2400 T^{3} + 265 p T^{4} - 16 p^{2} T^{5} + p^{3} T^{6} \) |
| 89 | $S_4\times C_2$ | \( 1 - 14 T + 263 T^{2} - 2308 T^{3} + 263 p T^{4} - 14 p^{2} T^{5} + p^{3} T^{6} \) |
| 97 | $S_4\times C_2$ | \( 1 - 18 T + 287 T^{2} - 3164 T^{3} + 287 p T^{4} - 18 p^{2} T^{5} + p^{3} T^{6} \) |
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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
−7.73805563555277294190762890568, −7.32351991414872381156112980752, −7.14926576789022755117576503508, −6.79616488451620252037944102444, −6.46875914920983328107036657766, −6.38042052374293757128324753759, −6.04478509660397442305761079050, −5.68747853841985563200841915448, −5.61106889293437592094915274678, −5.28770900701820343220553130160, −4.76993236148057957900794683435, −4.64250208422423109365138752872, −4.55102732382280850978697121132, −4.11262789093258421338317985499, −3.63886286975069824808964450458, −3.59922494799283353268932072001, −3.18755442266928233569086563225, −3.05462004037419664228231276523, −2.64401683932788002558649523917, −2.11046612306839665115103320443, −2.01643884792078798423550004150, −1.99148164412279942276585769230, −1.09748708760152999777154399496, −1.02699083718961863911607764203, −0.934462652721636924577550105518,
0.934462652721636924577550105518, 1.02699083718961863911607764203, 1.09748708760152999777154399496, 1.99148164412279942276585769230, 2.01643884792078798423550004150, 2.11046612306839665115103320443, 2.64401683932788002558649523917, 3.05462004037419664228231276523, 3.18755442266928233569086563225, 3.59922494799283353268932072001, 3.63886286975069824808964450458, 4.11262789093258421338317985499, 4.55102732382280850978697121132, 4.64250208422423109365138752872, 4.76993236148057957900794683435, 5.28770900701820343220553130160, 5.61106889293437592094915274678, 5.68747853841985563200841915448, 6.04478509660397442305761079050, 6.38042052374293757128324753759, 6.46875914920983328107036657766, 6.79616488451620252037944102444, 7.14926576789022755117576503508, 7.32351991414872381156112980752, 7.73805563555277294190762890568
