L(s) = 1 | − 8·3-s + 32·9-s − 4·13-s − 14·23-s − 6·25-s − 104·27-s − 4·31-s + 32·39-s + 36·41-s + 112·69-s + 44·71-s + 32·73-s + 48·75-s + 324·81-s + 32·93-s + 20·101-s − 128·117-s + 60·121-s − 288·123-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + ⋯ |
L(s) = 1 | − 4.61·3-s + 32/3·9-s − 1.10·13-s − 2.91·23-s − 6/5·25-s − 20.0·27-s − 0.718·31-s + 5.12·39-s + 5.62·41-s + 13.4·69-s + 5.22·71-s + 3.74·73-s + 5.54·75-s + 36·81-s + 3.31·93-s + 1.99·101-s − 11.8·117-s + 5.45·121-s − 25.9·123-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 5^{8} \cdot 23^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 5^{8} \cdot 23^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.3045138977\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.3045138977\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 + 6 T^{2} + 18 T^{4} + 6 p^{2} T^{6} + p^{4} T^{8} \) |
| 23 | \( 1 + 14 T + 98 T^{2} + 378 T^{3} + 1442 T^{4} + 378 p T^{5} + 98 p^{2} T^{6} + 14 p^{3} T^{7} + p^{4} T^{8} \) |
good | 3 | \( ( 1 + 2 T + 2 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{4} \) |
| 7 | \( 1 + 45 T^{4} + 788 T^{8} + 45 p^{4} T^{12} + p^{8} T^{16} \) |
| 11 | \( ( 1 - 30 T^{2} + 426 T^{4} - 30 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 13 | \( ( 1 + 2 T + 2 T^{2} - 14 T^{3} - 302 T^{4} - 14 p T^{5} + 2 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 17 | \( 1 + 69 T^{4} - 53260 T^{8} + 69 p^{4} T^{12} + p^{8} T^{16} \) |
| 19 | \( ( 1 + 18 T^{2} + 762 T^{4} + 18 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 29 | \( ( 1 - 35 T^{2} + 748 T^{4} - 35 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 31 | \( ( 1 + T - 30 T^{2} + p T^{3} + p^{2} T^{4} )^{4} \) |
| 37 | \( 1 + 17 p T^{4} - 1276220 T^{8} + 17 p^{5} T^{12} + p^{8} T^{16} \) |
| 41 | \( ( 1 - 9 T + 92 T^{2} - 9 p T^{3} + p^{2} T^{4} )^{4} \) |
| 43 | \( 1 - 3280 T^{4} + 9133438 T^{8} - 3280 p^{4} T^{12} + p^{8} T^{16} \) |
| 47 | \( ( 1 - 4274 T^{4} + p^{4} T^{8} )^{2} \) |
| 53 | \( 1 - 1387 T^{4} + 7377604 T^{8} - 1387 p^{4} T^{12} + p^{8} T^{16} \) |
| 59 | \( ( 1 - 215 T^{2} + 18508 T^{4} - 215 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 61 | \( ( 1 - 42 T^{2} + 5874 T^{4} - 42 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 67 | \( 1 - 12835 T^{4} + 81381988 T^{8} - 12835 p^{4} T^{12} + p^{8} T^{16} \) |
| 71 | \( ( 1 - 11 T + 162 T^{2} - 11 p T^{3} + p^{2} T^{4} )^{4} \) |
| 73 | \( ( 1 - 16 T + 128 T^{2} - 368 T^{3} - 1442 T^{4} - 368 p T^{5} + 128 p^{2} T^{6} - 16 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 79 | \( ( 1 + 82 T^{2} + 10842 T^{4} + 82 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 83 | \( 1 + 25277 T^{4} + 253565668 T^{8} + 25277 p^{4} T^{12} + p^{8} T^{16} \) |
| 89 | \( ( 1 - 122 T^{2} + 17554 T^{4} - 122 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 97 | \( 1 - 33280 T^{4} + 452590078 T^{8} - 33280 p^{4} T^{12} + p^{8} T^{16} \) |
show more | |
show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{16} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−4.92284170707532227815097926208, −4.89118462483169780890352076522, −4.69040410905823436668684182030, −4.59022329943581041641818492040, −4.42231288597151090460020337987, −4.16314779677423452288750425460, −4.00283528446624759119570319188, −3.94056031097119556921906432811, −3.84926422056568011156669168144, −3.72954674058311345377692518204, −3.71547888989345970484720092892, −3.38188272211060020690633611372, −3.05525358147149110767706438538, −3.01756611848481441741763060894, −2.45600868455045993838182632356, −2.20151039018333510742837497385, −2.18607161634898865016992946037, −2.18560690490802504663484974381, −2.05772276677572188733281530455, −1.82694397372285427955597590784, −1.34343477883820432084415182863, −1.02291089505521127061771939610, −0.59650957381359179539018029460, −0.51070447740054458583285326032, −0.39547779081891983190936324284,
0.39547779081891983190936324284, 0.51070447740054458583285326032, 0.59650957381359179539018029460, 1.02291089505521127061771939610, 1.34343477883820432084415182863, 1.82694397372285427955597590784, 2.05772276677572188733281530455, 2.18560690490802504663484974381, 2.18607161634898865016992946037, 2.20151039018333510742837497385, 2.45600868455045993838182632356, 3.01756611848481441741763060894, 3.05525358147149110767706438538, 3.38188272211060020690633611372, 3.71547888989345970484720092892, 3.72954674058311345377692518204, 3.84926422056568011156669168144, 3.94056031097119556921906432811, 4.00283528446624759119570319188, 4.16314779677423452288750425460, 4.42231288597151090460020337987, 4.59022329943581041641818492040, 4.69040410905823436668684182030, 4.89118462483169780890352076522, 4.92284170707532227815097926208
Plot not available for L-functions of degree greater than 10.