| L(s) = 1 | + 3-s − 2·5-s + 3·7-s + 2·11-s − 13-s − 2·15-s + 9·17-s − 3·19-s + 3·21-s + 11·23-s − 4·25-s − 5·27-s + 5·29-s + 12·31-s + 2·33-s − 6·35-s + 8·37-s − 39-s − 10·41-s − 8·43-s + 16·47-s − 5·49-s + 9·51-s + 13·53-s − 4·55-s − 3·57-s − 17·59-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.894·5-s + 1.13·7-s + 0.603·11-s − 0.277·13-s − 0.516·15-s + 2.18·17-s − 0.688·19-s + 0.654·21-s + 2.29·23-s − 4/5·25-s − 0.962·27-s + 0.928·29-s + 2.15·31-s + 0.348·33-s − 1.01·35-s + 1.31·37-s − 0.160·39-s − 1.56·41-s − 1.21·43-s + 2.33·47-s − 5/7·49-s + 1.26·51-s + 1.78·53-s − 0.539·55-s − 0.397·57-s − 2.21·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 19^{3}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 19^{3}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(7.687621657\) |
| \(L(\frac12)\) |
\(\approx\) |
\(7.687621657\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 19 | $C_1$ | \( ( 1 + T )^{3} \) |
| good | 3 | $S_4\times C_2$ | \( 1 - T + T^{2} + 4 T^{3} + p T^{4} - p^{2} T^{5} + p^{3} T^{6} \) |
| 5 | $S_4\times C_2$ | \( 1 + 2 T + 8 T^{2} + 22 T^{3} + 8 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 7 | $S_4\times C_2$ | \( 1 - 3 T + 2 p T^{2} - 37 T^{3} + 2 p^{2} T^{4} - 3 p^{2} T^{5} + p^{3} T^{6} \) |
| 11 | $S_4\times C_2$ | \( 1 - 2 T + 2 p T^{2} - 24 T^{3} + 2 p^{2} T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 13 | $S_4\times C_2$ | \( 1 + T + 27 T^{2} + 34 T^{3} + 27 p T^{4} + p^{2} T^{5} + p^{3} T^{6} \) |
| 17 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{3} \) |
| 23 | $S_4\times C_2$ | \( 1 - 11 T + 85 T^{2} - 426 T^{3} + 85 p T^{4} - 11 p^{2} T^{5} + p^{3} T^{6} \) |
| 29 | $S_4\times C_2$ | \( 1 - 5 T + 35 T^{2} - 16 T^{3} + 35 p T^{4} - 5 p^{2} T^{5} + p^{3} T^{6} \) |
| 31 | $S_4\times C_2$ | \( 1 - 12 T + 131 T^{2} - 764 T^{3} + 131 p T^{4} - 12 p^{2} T^{5} + p^{3} T^{6} \) |
| 37 | $S_4\times C_2$ | \( 1 - 8 T + 101 T^{2} - 492 T^{3} + 101 p T^{4} - 8 p^{2} T^{5} + p^{3} T^{6} \) |
| 41 | $S_4\times C_2$ | \( 1 + 10 T + 101 T^{2} + 620 T^{3} + 101 p T^{4} + 10 p^{2} T^{5} + p^{3} T^{6} \) |
| 43 | $S_4\times C_2$ | \( 1 + 8 T + 70 T^{2} + 338 T^{3} + 70 p T^{4} + 8 p^{2} T^{5} + p^{3} T^{6} \) |
| 47 | $S_4\times C_2$ | \( 1 - 16 T + 218 T^{2} - 1604 T^{3} + 218 p T^{4} - 16 p^{2} T^{5} + p^{3} T^{6} \) |
| 53 | $S_4\times C_2$ | \( 1 - 13 T + 207 T^{2} - 1416 T^{3} + 207 p T^{4} - 13 p^{2} T^{5} + p^{3} T^{6} \) |
| 59 | $S_4\times C_2$ | \( 1 + 17 T + 261 T^{2} + 2106 T^{3} + 261 p T^{4} + 17 p^{2} T^{5} + p^{3} T^{6} \) |
| 61 | $S_4\times C_2$ | \( 1 - 14 T + 212 T^{2} - 1566 T^{3} + 212 p T^{4} - 14 p^{2} T^{5} + p^{3} T^{6} \) |
| 67 | $S_4\times C_2$ | \( 1 - 29 T + 473 T^{2} - 4716 T^{3} + 473 p T^{4} - 29 p^{2} T^{5} + p^{3} T^{6} \) |
| 71 | $S_4\times C_2$ | \( 1 - 2 T + 89 T^{2} - 484 T^{3} + 89 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 73 | $S_4\times C_2$ | \( 1 + 17 T - 34 T^{2} - 1827 T^{3} - 34 p T^{4} + 17 p^{2} T^{5} + p^{3} T^{6} \) |
| 79 | $S_4\times C_2$ | \( 1 - 12 T + 275 T^{2} - 1916 T^{3} + 275 p T^{4} - 12 p^{2} T^{5} + p^{3} T^{6} \) |
| 83 | $S_4\times C_2$ | \( 1 - 16 T + 93 T^{2} - 96 T^{3} + 93 p T^{4} - 16 p^{2} T^{5} + p^{3} T^{6} \) |
| 89 | $S_4\times C_2$ | \( 1 + 20 T + 345 T^{2} + 3540 T^{3} + 345 p T^{4} + 20 p^{2} T^{5} + p^{3} T^{6} \) |
| 97 | $S_4\times C_2$ | \( 1 - 22 T + 211 T^{2} - 1468 T^{3} + 211 p T^{4} - 22 p^{2} T^{5} + p^{3} T^{6} \) |
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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
−7.34458026569919930774377831514, −7.20069999566721537778080870363, −6.90401637131460618633038138903, −6.69719826403227500703218746990, −6.31865203417142560288911949713, −6.10249579817980838937953178119, −5.95934334447129486968326001659, −5.38064907240580567594579993384, −5.32931127150668125960678840357, −5.07877516975672097140250359860, −4.78658719079720539816019678470, −4.59345335561874419756347663129, −4.31336086373903142737721962229, −3.86416719515890887530746644319, −3.78904506404028199971961752491, −3.50063352934143846167223194904, −3.27174396748307340100559483805, −2.72720540639833783706018321456, −2.69665760729017619018860259171, −2.36331688479846291871965529044, −1.91245321568771744698548853411, −1.37852415059533913297329194488, −1.31342156598890867096820472828, −0.67892283394192525110886825071, −0.63910437410652768700339337648,
0.63910437410652768700339337648, 0.67892283394192525110886825071, 1.31342156598890867096820472828, 1.37852415059533913297329194488, 1.91245321568771744698548853411, 2.36331688479846291871965529044, 2.69665760729017619018860259171, 2.72720540639833783706018321456, 3.27174396748307340100559483805, 3.50063352934143846167223194904, 3.78904506404028199971961752491, 3.86416719515890887530746644319, 4.31336086373903142737721962229, 4.59345335561874419756347663129, 4.78658719079720539816019678470, 5.07877516975672097140250359860, 5.32931127150668125960678840357, 5.38064907240580567594579993384, 5.95934334447129486968326001659, 6.10249579817980838937953178119, 6.31865203417142560288911949713, 6.69719826403227500703218746990, 6.90401637131460618633038138903, 7.20069999566721537778080870363, 7.34458026569919930774377831514
