| L(s) = 1 | − 2·4-s + 3·16-s − 2·19-s − 25-s + 2·49-s + 4·61-s − 4·64-s + 4·76-s + 2·100-s − 2·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 2·169-s + 173-s + 179-s + 181-s + 191-s + 193-s − 4·196-s + 197-s + ⋯ |
| L(s) = 1 | − 2·4-s + 3·16-s − 2·19-s − 25-s + 2·49-s + 4·61-s − 4·64-s + 4·76-s + 2·100-s − 2·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 2·169-s + 173-s + 179-s + 181-s + 191-s + 193-s − 4·196-s + 197-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 731025 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 731025 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(0.4709855956\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4709855956\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | | \( 1 \) |
| 5 | $C_2$ | \( 1 + T^{2} \) |
| 19 | $C_1$ | \( ( 1 + T )^{2} \) |
| good | 2 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 7 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 11 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 29 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 31 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 37 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 41 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 43 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 47 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 59 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 61 | $C_1$ | \( ( 1 - T )^{4} \) |
| 67 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 71 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 73 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 79 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 83 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 89 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 97 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
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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
−10.39362992561294684367144717927, −9.938023508716114438615412539379, −9.935224915325706876146880644565, −9.233680695753324646036289475510, −8.844812329386031709516243281533, −8.622554972135854310651242063969, −8.159786214993360024317458354109, −7.896313259826873230734736719258, −7.23354116876202208422031411645, −6.71051172070228487336639076565, −6.16485284536393311714688905889, −5.52956726641953847845006122904, −5.43089577767191624653108792806, −4.72730467283073990681562354768, −4.14212187204367362594602075666, −4.04005117855751908584612703180, −3.51522965472445254476240749136, −2.59228109815596430842835535724, −1.90073524500712755725850802148, −0.73410684909916266227780964106,
0.73410684909916266227780964106, 1.90073524500712755725850802148, 2.59228109815596430842835535724, 3.51522965472445254476240749136, 4.04005117855751908584612703180, 4.14212187204367362594602075666, 4.72730467283073990681562354768, 5.43089577767191624653108792806, 5.52956726641953847845006122904, 6.16485284536393311714688905889, 6.71051172070228487336639076565, 7.23354116876202208422031411645, 7.896313259826873230734736719258, 8.159786214993360024317458354109, 8.622554972135854310651242063969, 8.844812329386031709516243281533, 9.233680695753324646036289475510, 9.935224915325706876146880644565, 9.938023508716114438615412539379, 10.39362992561294684367144717927
