| L(s) = 1 | − 2·3-s + 2·5-s + 5·9-s + 2·11-s + 20·13-s − 4·15-s − 10·17-s + 4·19-s − 4·23-s + 25-s − 10·27-s + 4·29-s − 12·31-s − 4·33-s + 4·37-s − 40·39-s + 8·41-s + 24·43-s + 10·45-s − 2·47-s + 20·51-s + 16·53-s + 4·55-s − 8·57-s − 4·59-s − 16·61-s + 40·65-s + ⋯ |
| L(s) = 1 | − 1.15·3-s + 0.894·5-s + 5/3·9-s + 0.603·11-s + 5.54·13-s − 1.03·15-s − 2.42·17-s + 0.917·19-s − 0.834·23-s + 1/5·25-s − 1.92·27-s + 0.742·29-s − 2.15·31-s − 0.696·33-s + 0.657·37-s − 6.40·39-s + 1.24·41-s + 3.65·43-s + 1.49·45-s − 0.291·47-s + 2.80·51-s + 2.19·53-s + 0.539·55-s − 1.05·57-s − 0.520·59-s − 2.04·61-s + 4.96·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 5^{4} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 5^{4} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.429963508\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.429963508\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( ( 1 - T + T^{2} )^{2} \) |
| 7 | | \( 1 \) |
| good | 3 | $D_4\times C_2$ | \( 1 + 2 T - T^{2} - 2 T^{3} + 4 T^{4} - 2 p T^{5} - p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 11 | $D_4\times C_2$ | \( 1 - 2 T - p T^{2} + 14 T^{3} + 60 T^{4} + 14 p T^{5} - p^{3} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 13 | $D_{4}$ | \( ( 1 - 10 T + 49 T^{2} - 10 p T^{3} + p^{2} T^{4} )^{2} \) |
| 17 | $D_4\times C_2$ | \( 1 + 10 T + 43 T^{2} + 230 T^{3} + 1260 T^{4} + 230 p T^{5} + 43 p^{2} T^{6} + 10 p^{3} T^{7} + p^{4} T^{8} \) |
| 19 | $D_4\times C_2$ | \( 1 - 4 T + 6 T^{2} + 112 T^{3} - 565 T^{4} + 112 p T^{5} + 6 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 23 | $D_4\times C_2$ | \( 1 + 4 T - 32 T^{2} + 8 T^{3} + 1407 T^{4} + 8 p T^{5} - 32 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 29 | $D_{4}$ | \( ( 1 - 2 T + 27 T^{2} - 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 31 | $D_4\times C_2$ | \( 1 + 12 T + 48 T^{2} + 408 T^{3} + 3791 T^{4} + 408 p T^{5} + 48 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} \) |
| 37 | $D_4\times C_2$ | \( 1 - 4 T - 60 T^{2} - 8 T^{3} + 3815 T^{4} - 8 p T^{5} - 60 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 41 | $D_{4}$ | \( ( 1 - 4 T + 84 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 43 | $D_{4}$ | \( ( 1 - 12 T + 90 T^{2} - 12 p T^{3} + p^{2} T^{4} )^{2} \) |
| 47 | $D_4\times C_2$ | \( 1 + 2 T + 7 T^{2} - 194 T^{3} - 2388 T^{4} - 194 p T^{5} + 7 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 53 | $D_4\times C_2$ | \( 1 - 16 T + 104 T^{2} - 736 T^{3} + 6727 T^{4} - 736 p T^{5} + 104 p^{2} T^{6} - 16 p^{3} T^{7} + p^{4} T^{8} \) |
| 59 | $D_4\times C_2$ | \( 1 + 4 T - 104 T^{2} + 8 T^{3} + 9975 T^{4} + 8 p T^{5} - 104 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 61 | $D_4\times C_2$ | \( 1 + 16 T + 78 T^{2} + 896 T^{3} + 12347 T^{4} + 896 p T^{5} + 78 p^{2} T^{6} + 16 p^{3} T^{7} + p^{4} T^{8} \) |
| 67 | $D_4\times C_2$ | \( 1 - 8 T - 36 T^{2} + 272 T^{3} + 1223 T^{4} + 272 p T^{5} - 36 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} \) |
| 71 | $D_{4}$ | \( ( 1 + 4 T + 74 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 73 | $D_4\times C_2$ | \( 1 + 16 T + 54 T^{2} + 896 T^{3} + 16787 T^{4} + 896 p T^{5} + 54 p^{2} T^{6} + 16 p^{3} T^{7} + p^{4} T^{8} \) |
| 79 | $D_4\times C_2$ | \( 1 - 2 T + 45 T^{2} + 398 T^{3} - 4876 T^{4} + 398 p T^{5} + 45 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 83 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{4} \) |
| 89 | $C_2^3$ | \( 1 + 110 T^{2} + 4179 T^{4} + 110 p^{2} T^{6} + p^{4} T^{8} \) |
| 97 | $D_{4}$ | \( ( 1 - 6 T + 41 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 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
−7.16762172490445618376724569834, −6.73495918881274434226356921658, −6.60486821115730914193114835539, −6.46001693798905485993066490741, −6.05530963843593604773144146971, −5.98681169556613566624862287789, −5.86548568987095842157864987937, −5.76714170782498258072765011702, −5.68333913634415324527727453250, −5.27154887843716831362354285479, −4.68056568379224875526112334212, −4.44545833994665623062774904478, −4.43548084738216714837978559823, −4.01235141907713258342592950034, −3.90886430637287427107357005214, −3.74586305369303139155045424295, −3.51943150448540369835050929002, −3.05302419772051504201289514073, −2.55069355491189466451625478203, −2.41997427598441616360141008052, −1.72149879030093262344547115005, −1.63595170745438035974045341450, −1.31757867481311323134413049381, −1.05858090621102130325001002481, −0.59063616646384276336519801481,
0.59063616646384276336519801481, 1.05858090621102130325001002481, 1.31757867481311323134413049381, 1.63595170745438035974045341450, 1.72149879030093262344547115005, 2.41997427598441616360141008052, 2.55069355491189466451625478203, 3.05302419772051504201289514073, 3.51943150448540369835050929002, 3.74586305369303139155045424295, 3.90886430637287427107357005214, 4.01235141907713258342592950034, 4.43548084738216714837978559823, 4.44545833994665623062774904478, 4.68056568379224875526112334212, 5.27154887843716831362354285479, 5.68333913634415324527727453250, 5.76714170782498258072765011702, 5.86548568987095842157864987937, 5.98681169556613566624862287789, 6.05530963843593604773144146971, 6.46001693798905485993066490741, 6.60486821115730914193114835539, 6.73495918881274434226356921658, 7.16762172490445618376724569834
