| L(s) = 1 | + 2-s + 5-s + 7-s − 8-s + 10-s + 2·11-s − 3·13-s + 14-s − 16-s + 2·17-s + 4·19-s + 2·22-s + 2·23-s + 5·25-s − 3·26-s + 7·29-s − 6·31-s + 2·34-s + 35-s − 14·37-s + 4·38-s − 40-s + 6·41-s + 4·43-s + 2·46-s + 6·47-s + 5·50-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.447·5-s + 0.377·7-s − 0.353·8-s + 0.316·10-s + 0.603·11-s − 0.832·13-s + 0.267·14-s − 1/4·16-s + 0.485·17-s + 0.917·19-s + 0.426·22-s + 0.417·23-s + 25-s − 0.588·26-s + 1.29·29-s − 1.07·31-s + 0.342·34-s + 0.169·35-s − 2.30·37-s + 0.648·38-s − 0.158·40-s + 0.937·41-s + 0.609·43-s + 0.294·46-s + 0.875·47-s + 0.707·50-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1285956 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1285956 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.622017085\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.622017085\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 - T + T^{2} \) |
| 3 | | \( 1 \) |
| 7 | $C_2$ | \( 1 - T + T^{2} \) |
| good | 5 | $C_2^2$ | \( 1 - T - 4 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 2 T - 7 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 + 3 T - 4 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 2 T - 19 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 7 T + 20 T^{2} - 7 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2^2$ | \( 1 + 6 T + 5 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2$ | \( ( 1 + 7 T + p T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 - 6 T - 5 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 - 4 T - 27 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 6 T - 11 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 - 10 T + 41 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 9 T + 20 T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 + 10 T + 33 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 + 11 T + p T^{2} )^{2} \) |
| 79 | $C_2^2$ | \( 1 - 6 T - 43 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2^2$ | \( 1 + 10 T + 17 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + 15 T + p T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 - 2 T - 93 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.891276605657681297051646313462, −9.849925585394424269974296993712, −9.002205689416305844392476375662, −8.892779965060065374146509305820, −8.553063722587105000292773192718, −7.966366491845029999554553336174, −7.24782868106683910884718979818, −7.06122623739023385344683633553, −6.90914957514134103574809424616, −6.03713124097115526623001182144, −5.65252103462601336191879523824, −5.28420150971968308719983300573, −5.03412561882234343657160946940, −4.19942461464256383474112257531, −4.15088848117274565417116831695, −3.25466925627715361937013405711, −2.92427982104566816492156765746, −2.29161131267906432441879891888, −1.52555650207311047170545100730, −0.794205433141935869470282525048,
0.794205433141935869470282525048, 1.52555650207311047170545100730, 2.29161131267906432441879891888, 2.92427982104566816492156765746, 3.25466925627715361937013405711, 4.15088848117274565417116831695, 4.19942461464256383474112257531, 5.03412561882234343657160946940, 5.28420150971968308719983300573, 5.65252103462601336191879523824, 6.03713124097115526623001182144, 6.90914957514134103574809424616, 7.06122623739023385344683633553, 7.24782868106683910884718979818, 7.966366491845029999554553336174, 8.553063722587105000292773192718, 8.892779965060065374146509305820, 9.002205689416305844392476375662, 9.849925585394424269974296993712, 9.891276605657681297051646313462
