| L(s) = 1 | + 2·2-s − 6·3-s + 4-s + 3·5-s − 12·6-s − 3·7-s − 2·8-s + 21·9-s + 6·10-s + 5·11-s − 6·12-s − 6·14-s − 18·15-s − 4·16-s + 17-s + 42·18-s − 15·19-s + 3·20-s + 18·21-s + 10·22-s + 3·23-s + 12·24-s − 54·27-s − 3·28-s + 4·29-s − 36·30-s + 13·31-s + ⋯ |
| L(s) = 1 | + 1.41·2-s − 3.46·3-s + 1/2·4-s + 1.34·5-s − 4.89·6-s − 1.13·7-s − 0.707·8-s + 7·9-s + 1.89·10-s + 1.50·11-s − 1.73·12-s − 1.60·14-s − 4.64·15-s − 16-s + 0.242·17-s + 9.89·18-s − 3.44·19-s + 0.670·20-s + 3.92·21-s + 2.13·22-s + 0.625·23-s + 2.44·24-s − 10.3·27-s − 0.566·28-s + 0.742·29-s − 6.57·30-s + 2.33·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{4} \cdot 7^{4} \cdot 11^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{4} \cdot 7^{4} \cdot 11^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.9886748954\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9886748954\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( ( 1 - T + T^{2} )^{2} \) |
| 3 | $C_2$ | \( ( 1 + p T + p T^{2} )^{2} \) |
| 7 | $C_2^2$ | \( 1 + 3 T + 2 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 5 T + 14 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| good | 5 | $C_2$$\times$$C_2^2$ | \( ( 1 - T + p T^{2} )^{2}( 1 - T - 4 T^{2} - p T^{3} + p^{2} T^{4} ) \) |
| 13 | $D_4\times C_2$ | \( 1 - 8 T^{2} + 126 T^{4} - 8 p^{2} T^{6} + p^{4} T^{8} \) |
| 17 | $D_4\times C_2$ | \( 1 - T - 19 T^{2} + 14 T^{3} + 94 T^{4} + 14 p T^{5} - 19 p^{2} T^{6} - p^{3} T^{7} + p^{4} T^{8} \) |
| 19 | $D_4\times C_2$ | \( 1 + 15 T + 127 T^{2} + 780 T^{3} + 3768 T^{4} + 780 p T^{5} + 127 p^{2} T^{6} + 15 p^{3} T^{7} + p^{4} T^{8} \) |
| 23 | $D_4\times C_2$ | \( 1 - 3 T + 7 T^{2} - 12 T^{3} - 444 T^{4} - 12 p T^{5} + 7 p^{2} T^{6} - 3 p^{3} T^{7} + p^{4} T^{8} \) |
| 29 | $C_2$ | \( ( 1 - T + p T^{2} )^{4} \) |
| 31 | $D_4\times C_2$ | \( 1 - 13 T + 79 T^{2} - 364 T^{3} + 1900 T^{4} - 364 p T^{5} + 79 p^{2} T^{6} - 13 p^{3} T^{7} + p^{4} T^{8} \) |
| 37 | $D_4\times C_2$ | \( 1 - 7 T - 23 T^{2} + 14 T^{3} + 2002 T^{4} + 14 p T^{5} - 23 p^{2} T^{6} - 7 p^{3} T^{7} + p^{4} T^{8} \) |
| 41 | $D_{4}$ | \( ( 1 - 2 T + 26 T^{2} - 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 43 | $D_4\times C_2$ | \( 1 - 41 T^{2} + 2964 T^{4} - 41 p^{2} T^{6} + p^{4} T^{8} \) |
| 47 | $D_4\times C_2$ | \( 1 + 15 T + 145 T^{2} + 1050 T^{3} + 6216 T^{4} + 1050 p T^{5} + 145 p^{2} T^{6} + 15 p^{3} T^{7} + p^{4} T^{8} \) |
| 53 | $D_4\times C_2$ | \( 1 - 9 T + 21 T^{2} + 54 T^{3} - 1342 T^{4} + 54 p T^{5} + 21 p^{2} T^{6} - 9 p^{3} T^{7} + p^{4} T^{8} \) |
| 59 | $C_2^2$ | \( ( 1 - 3 T + 62 T^{2} - 3 p T^{3} + p^{2} T^{4} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - T + p T^{2} )^{2}( 1 + 13 T + p T^{2} )^{2} \) |
| 67 | $D_4\times C_2$ | \( 1 - 2 T - 74 T^{2} + 112 T^{3} + 1327 T^{4} + 112 p T^{5} - 74 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 71 | $D_4\times C_2$ | \( 1 - 153 T^{2} + 14780 T^{4} - 153 p^{2} T^{6} + p^{4} T^{8} \) |
| 73 | $D_4\times C_2$ | \( 1 - 12 T + 130 T^{2} - 984 T^{3} + 4899 T^{4} - 984 p T^{5} + 130 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8} \) |
| 79 | $D_4\times C_2$ | \( 1 + 15 T + 247 T^{2} + 2580 T^{3} + 29268 T^{4} + 2580 p T^{5} + 247 p^{2} T^{6} + 15 p^{3} T^{7} + p^{4} T^{8} \) |
| 83 | $D_{4}$ | \( ( 1 - 13 T + 194 T^{2} - 13 p T^{3} + p^{2} T^{4} )^{2} \) |
| 89 | $D_4\times C_2$ | \( 1 + 6 T + 174 T^{2} + 972 T^{3} + 19391 T^{4} + 972 p T^{5} + 174 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 97 | $D_{4}$ | \( ( 1 + 12 T + 173 T^{2} + 12 p T^{3} + p^{2} T^{4} )^{2} \) |
| show more | | |
| show less | | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.071782047232941215529693891259, −7.64243499821712518522144808683, −6.95987871615479347008302648112, −6.71718815558134730133236770552, −6.68672756776638948622203020913, −6.65780184701942052251785932214, −6.47570865736353871190525546865, −6.18634079693378358559496130663, −6.00725779775747951782304529813, −5.67043403645455554951024237999, −5.53703346840373901460473597738, −5.51181859897001382206390839456, −4.90443302478680409877222450870, −4.62012884148821899606724632983, −4.54965996642917334893247881179, −4.20777083617075768569707484115, −4.09686518319191644436420937094, −3.93671163128592824140193284173, −3.20215504992628407221056189612, −2.87288459780854187868838357520, −2.53458499637883610415310211941, −1.86754804333366106405253969971, −1.61402658895754801508188353261, −0.957133372024537941891006190336, −0.43024360843908035024280316747,
0.43024360843908035024280316747, 0.957133372024537941891006190336, 1.61402658895754801508188353261, 1.86754804333366106405253969971, 2.53458499637883610415310211941, 2.87288459780854187868838357520, 3.20215504992628407221056189612, 3.93671163128592824140193284173, 4.09686518319191644436420937094, 4.20777083617075768569707484115, 4.54965996642917334893247881179, 4.62012884148821899606724632983, 4.90443302478680409877222450870, 5.51181859897001382206390839456, 5.53703346840373901460473597738, 5.67043403645455554951024237999, 6.00725779775747951782304529813, 6.18634079693378358559496130663, 6.47570865736353871190525546865, 6.65780184701942052251785932214, 6.68672756776638948622203020913, 6.71718815558134730133236770552, 6.95987871615479347008302648112, 7.64243499821712518522144808683, 8.071782047232941215529693891259
