L(s) = 1 | − 4-s + 16-s + 4·19-s + 2·31-s + 2·49-s + 2·61-s − 2·64-s − 4·76-s − 2·79-s + 4·109-s − 2·121-s − 2·124-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 + ⋯ |
L(s) = 1 | − 4-s + 16-s + 4·19-s + 2·31-s + 2·49-s + 2·61-s − 2·64-s − 4·76-s − 2·79-s + 4·109-s − 2·121-s − 2·124-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 + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{16} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{16} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.375176662\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.375176662\) |
\(L(1)\) |
|
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 | | \( 1 \) |
good | 2 | $C_2$$\times$$C_2^2$ | \( ( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \) |
| 7 | $C_2^2$ | \( ( 1 - T^{2} + T^{4} )^{2} \) |
| 11 | $C_2$ | \( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( ( 1 - T^{2} + T^{4} )^{2} \) |
| 17 | $C_2^2$ | \( ( 1 - T^{2} + T^{4} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - T + T^{2} )^{4} \) |
| 23 | $C_2$$\times$$C_2^2$ | \( ( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \) |
| 29 | $C_2$ | \( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \) |
| 31 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{4}( 1 + T + T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 41 | $C_2$ | \( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( ( 1 - T^{2} + T^{4} )^{2} \) |
| 47 | $C_2^2$ | \( ( 1 - T^{2} + T^{4} )^{2} \) |
| 53 | $C_2^2$ | \( ( 1 - T^{2} + T^{4} )^{2} \) |
| 59 | $C_2$ | \( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \) |
| 61 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{4}( 1 + T + T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( ( 1 - T^{2} + T^{4} )^{2} \) |
| 71 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{4}( 1 + T )^{4} \) |
| 73 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 79 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{4}( 1 - T + T^{2} )^{2} \) |
| 83 | $C_2$$\times$$C_2^2$ | \( ( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \) |
| 89 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{4}( 1 + T )^{4} \) |
| 97 | $C_2^2$ | \( ( 1 - T^{2} + T^{4} )^{2} \) |
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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
−6.88104858651300994514661298742, −6.41247542612574389861234128499, −6.07955915592130985422612419225, −6.06028525691658715527193952385, −5.91177179035889751102159309482, −5.53790045439090630797049024542, −5.41546441004578799070022287827, −5.22382278892486172834765675987, −5.12979085574737022115734054580, −4.76106910382948079635209442445, −4.45715624614404614809135921284, −4.38599537733489761626788181243, −4.37285045112079370579871923867, −3.61861895121589991387926485869, −3.57076606916853206557710788218, −3.54517670398687785612810433522, −3.28404949262021698157272904817, −2.78742674956118604913233033635, −2.78490642555186432135423112300, −2.42929120577766331274259350851, −2.15575717556613149128907223841, −1.59317375478164994171015730582, −1.13816103219755869098658394721, −1.10275190080177178849316301460, −0.796436407443435296753350459483,
0.796436407443435296753350459483, 1.10275190080177178849316301460, 1.13816103219755869098658394721, 1.59317375478164994171015730582, 2.15575717556613149128907223841, 2.42929120577766331274259350851, 2.78490642555186432135423112300, 2.78742674956118604913233033635, 3.28404949262021698157272904817, 3.54517670398687785612810433522, 3.57076606916853206557710788218, 3.61861895121589991387926485869, 4.37285045112079370579871923867, 4.38599537733489761626788181243, 4.45715624614404614809135921284, 4.76106910382948079635209442445, 5.12979085574737022115734054580, 5.22382278892486172834765675987, 5.41546441004578799070022287827, 5.53790045439090630797049024542, 5.91177179035889751102159309482, 6.06028525691658715527193952385, 6.07955915592130985422612419225, 6.41247542612574389861234128499, 6.88104858651300994514661298742