| L(s) = 1 | + 4-s + 4·9-s + 4·36-s − 2·49-s − 64-s + 4·71-s − 4·79-s + 10·81-s − 2·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 4·169-s + 173-s + 179-s + 181-s + 191-s + 193-s − 2·196-s + 197-s + 199-s + ⋯ |
| L(s) = 1 | + 4-s + 4·9-s + 4·36-s − 2·49-s − 64-s + 4·71-s − 4·79-s + 10·81-s − 2·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 4·169-s + 173-s + 179-s + 181-s + 191-s + 193-s − 2·196-s + 197-s + 199-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 5^{8} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 5^{8} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(2.219330537\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.219330537\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 5 | | \( 1 \) |
| 7 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| good | 3 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{4}( 1 + T )^{4} \) |
| 11 | $C_2$ | \( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \) |
| 13 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{4}( 1 + T )^{4} \) |
| 17 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 19 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 23 | $C_2^2$ | \( ( 1 - T^{2} + T^{4} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \) |
| 31 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{4}( 1 + T )^{4} \) |
| 37 | $C_2^2$ | \( ( 1 - T^{2} + T^{4} )^{2} \) |
| 41 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{4}( 1 + T )^{4} \) |
| 43 | $C_2^2$ | \( ( 1 - T^{2} + T^{4} )^{2} \) |
| 47 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 53 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 59 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 61 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 67 | $C_2^2$ | \( ( 1 - T^{2} + T^{4} )^{2} \) |
| 71 | $C_2$ | \( ( 1 - T + T^{2} )^{4} \) |
| 73 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 79 | $C_2$ | \( ( 1 + T + T^{2} )^{4} \) |
| 83 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{4}( 1 + T )^{4} \) |
| 89 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{4}( 1 + T )^{4} \) |
| 97 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
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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
−6.95326472785480355241422688688, −6.86453488460322438690075566461, −6.70527757626231824758033501349, −6.50277314686949515194335874459, −6.44221876745650083029023119916, −5.95825078169335581832940843414, −5.76213838251622997730402030452, −5.57637640297452060270368011031, −5.23719777479881426953485330547, −4.85128989002180292285306507561, −4.68097393040551552482862211262, −4.65668643985632725499620215615, −4.44715184985955166488361882859, −4.11814511370719721310756726033, −3.78360694597269156595348377952, −3.67399747289231948260723593332, −3.33390983473758282201081445631, −3.32281008630357227150847568278, −2.49039640507032635732273873908, −2.42266313709867637572671166885, −2.31106944723953951983158795480, −1.77614059708378109395401590793, −1.52034457622724447022130108464, −1.23622711660360511502588434856, −1.13550396773616017793949129346,
1.13550396773616017793949129346, 1.23622711660360511502588434856, 1.52034457622724447022130108464, 1.77614059708378109395401590793, 2.31106944723953951983158795480, 2.42266313709867637572671166885, 2.49039640507032635732273873908, 3.32281008630357227150847568278, 3.33390983473758282201081445631, 3.67399747289231948260723593332, 3.78360694597269156595348377952, 4.11814511370719721310756726033, 4.44715184985955166488361882859, 4.65668643985632725499620215615, 4.68097393040551552482862211262, 4.85128989002180292285306507561, 5.23719777479881426953485330547, 5.57637640297452060270368011031, 5.76213838251622997730402030452, 5.95825078169335581832940843414, 6.44221876745650083029023119916, 6.50277314686949515194335874459, 6.70527757626231824758033501349, 6.86453488460322438690075566461, 6.95326472785480355241422688688
