| L(s) = 1 | − 2·5-s + 4·7-s − 4·11-s + 4·13-s − 16·17-s + 8·19-s − 4·23-s + 25-s + 10·29-s − 4·31-s − 8·35-s + 6·41-s − 16·43-s − 4·47-s + 15·49-s − 24·53-s + 8·55-s − 8·59-s + 10·61-s − 8·65-s + 16·67-s + 24·71-s − 16·77-s + 24·79-s + 16·83-s + 32·85-s + 12·89-s + ⋯ |
| L(s) = 1 | − 0.894·5-s + 1.51·7-s − 1.20·11-s + 1.10·13-s − 3.88·17-s + 1.83·19-s − 0.834·23-s + 1/5·25-s + 1.85·29-s − 0.718·31-s − 1.35·35-s + 0.937·41-s − 2.43·43-s − 0.583·47-s + 15/7·49-s − 3.29·53-s + 1.07·55-s − 1.04·59-s + 1.28·61-s − 0.992·65-s + 1.95·67-s + 2.84·71-s − 1.82·77-s + 2.70·79-s + 1.75·83-s + 3.47·85-s + 1.27·89-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{12} \cdot 5^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{12} \cdot 5^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.1790747144\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1790747144\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 5 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| good | 7 | $D_4\times C_2$ | \( 1 - 4 T + T^{2} - 4 T^{3} + 64 T^{4} - 4 p T^{5} + p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 11 | $D_4\times C_2$ | \( 1 + 4 T + 2 T^{2} - 32 T^{3} - 101 T^{4} - 32 p T^{5} + 2 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 13 | $D_4\times C_2$ | \( 1 - 4 T - 2 T^{2} + 32 T^{3} - 53 T^{4} + 32 p T^{5} - 2 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 17 | $D_{4}$ | \( ( 1 + 8 T + 38 T^{2} + 8 p T^{3} + p^{2} T^{4} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{4} \) |
| 23 | $D_4\times C_2$ | \( 1 + 4 T - 31 T^{2} + 4 T^{3} + 1312 T^{4} + 4 p T^{5} - 31 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 29 | $C_2$$\times$$C_2^2$ | \( ( 1 - 10 T + p T^{2} )^{2}( 1 + 10 T + 71 T^{2} + 10 p T^{3} + p^{2} T^{4} ) \) |
| 31 | $C_2^2$ | \( ( 1 + 2 T - 27 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 37 | $C_2^2$ | \( ( 1 - 34 T^{2} + p^{2} T^{4} )^{2} \) |
| 41 | $D_4\times C_2$ | \( 1 - 6 T - 7 T^{2} + 234 T^{3} - 1308 T^{4} + 234 p T^{5} - 7 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 43 | $D_4\times C_2$ | \( 1 + 16 T + 118 T^{2} + 832 T^{3} + 6187 T^{4} + 832 p T^{5} + 118 p^{2} T^{6} + 16 p^{3} T^{7} + p^{4} T^{8} \) |
| 47 | $D_4\times C_2$ | \( 1 + 4 T - 79 T^{2} + 4 T^{3} + 6064 T^{4} + 4 p T^{5} - 79 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 53 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{4} \) |
| 59 | $D_4\times C_2$ | \( 1 + 8 T + 38 T^{2} - 736 T^{3} - 6581 T^{4} - 736 p T^{5} + 38 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} \) |
| 61 | $D_4\times C_2$ | \( 1 - 10 T - 35 T^{2} - 130 T^{3} + 8404 T^{4} - 130 p T^{5} - 35 p^{2} T^{6} - 10 p^{3} T^{7} + p^{4} T^{8} \) |
| 67 | $D_4\times C_2$ | \( 1 - 16 T + 61 T^{2} - 976 T^{3} + 16384 T^{4} - 976 p T^{5} + 61 p^{2} T^{6} - 16 p^{3} T^{7} + p^{4} T^{8} \) |
| 71 | $D_{4}$ | \( ( 1 - 12 T + 166 T^{2} - 12 p T^{3} + p^{2} T^{4} )^{2} \) |
| 73 | $C_2^2$ | \( ( 1 + 98 T^{2} + p^{2} T^{4} )^{2} \) |
| 79 | $D_4\times C_2$ | \( 1 - 24 T + 286 T^{2} - 3168 T^{3} + 32355 T^{4} - 3168 p T^{5} + 286 p^{2} T^{6} - 24 p^{3} T^{7} + p^{4} T^{8} \) |
| 83 | $D_4\times C_2$ | \( 1 - 16 T + 53 T^{2} - 592 T^{3} + 13072 T^{4} - 592 p T^{5} + 53 p^{2} T^{6} - 16 p^{3} T^{7} + p^{4} T^{8} \) |
| 89 | $D_{4}$ | \( ( 1 - 6 T + 139 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 97 | $D_4\times C_2$ | \( 1 - 4 T - 134 T^{2} + 176 T^{3} + 11539 T^{4} + 176 p T^{5} - 134 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−6.43680580679924451104690349274, −6.42725159080312343044274175579, −6.19551012861373935479170853890, −5.88774781281098208727645206792, −5.31563525225271243879866977467, −5.24528185166705126556254737932, −5.14242167740956203890334305998, −4.98894863731014607345378771328, −4.85742516698642617113826521396, −4.62176582837676369075936223040, −4.25053322852681914746467729030, −4.05981116256698429181254831358, −4.00643124339954078042677038716, −3.58959363533982929638887635145, −3.33876197036055713726498017737, −3.33488503960487398422299693220, −2.73946666455999579449883434709, −2.57425407281440385937809878161, −2.30373378258580713665776380240, −2.06816402390783266582383411502, −1.82803720839014302680172813056, −1.55582838615027315705183593673, −0.907493022977264738873338658089, −0.891179820985886843111427235684, −0.07793196059935338284176889426,
0.07793196059935338284176889426, 0.891179820985886843111427235684, 0.907493022977264738873338658089, 1.55582838615027315705183593673, 1.82803720839014302680172813056, 2.06816402390783266582383411502, 2.30373378258580713665776380240, 2.57425407281440385937809878161, 2.73946666455999579449883434709, 3.33488503960487398422299693220, 3.33876197036055713726498017737, 3.58959363533982929638887635145, 4.00643124339954078042677038716, 4.05981116256698429181254831358, 4.25053322852681914746467729030, 4.62176582837676369075936223040, 4.85742516698642617113826521396, 4.98894863731014607345378771328, 5.14242167740956203890334305998, 5.24528185166705126556254737932, 5.31563525225271243879866977467, 5.88774781281098208727645206792, 6.19551012861373935479170853890, 6.42725159080312343044274175579, 6.43680580679924451104690349274
