| L(s) = 1 | − 2·4-s − 4·9-s + 3·16-s + 8·36-s − 4·64-s + 10·81-s + 4·121-s + 127-s + 131-s + 137-s + 139-s − 12·144-s + 149-s + 151-s + 157-s + 163-s + 167-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + 227-s + ⋯ |
| L(s) = 1 | − 2·4-s − 4·9-s + 3·16-s + 8·36-s − 4·64-s + 10·81-s + 4·121-s + 127-s + 131-s + 137-s + 139-s − 12·144-s + 149-s + 151-s + 157-s + 163-s + 167-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + 227-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 5^{4} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 5^{4} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(0.2541243197\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2541243197\) |
| \(L(1)\) |
|
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 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 5 | $C_2^2$ | \( 1 + T^{4} \) |
| 7 | | \( 1 \) |
| good | 3 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 11 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{4}( 1 + T )^{4} \) |
| 13 | $C_2^2$ | \( ( 1 + T^{4} )^{2} \) |
| 17 | $C_2^2$ | \( ( 1 + T^{4} )^{2} \) |
| 19 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{4}( 1 + T )^{4} \) |
| 23 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 29 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 31 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{4}( 1 + T )^{4} \) |
| 37 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 41 | $C_2^2$ | \( ( 1 + T^{4} )^{2} \) |
| 43 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 47 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 53 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{4}( 1 + T )^{4} \) |
| 59 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{4}( 1 + T )^{4} \) |
| 61 | $C_2^2$ | \( ( 1 + T^{4} )^{2} \) |
| 67 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 71 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{4}( 1 + T )^{4} \) |
| 73 | $C_2^2$ | \( ( 1 + T^{4} )^{2} \) |
| 79 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{4}( 1 + T )^{4} \) |
| 83 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 89 | $C_2^2$ | \( ( 1 + T^{4} )^{2} \) |
| 97 | $C_2^2$ | \( ( 1 + T^{4} )^{2} \) |
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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
−7.68497850518753325857033312683, −7.03866921314571519093449085626, −6.89604458435717238916698407724, −6.84961743495114197441092533626, −6.11862045358734173184993109672, −6.11804106913666178614538117482, −6.01430082940369843112638691450, −5.79465799480767257051592905319, −5.45404758597821470960295222444, −5.32981745365429528355255310106, −5.30304970252868633953811365137, −4.74815458656768011716886465042, −4.69475435301017811937614412610, −4.48495158731571501877475388209, −4.01903251288157073759077382382, −3.83947385823145028259015544178, −3.46198149452825133429845054148, −3.19758381824163121236584394200, −3.02755718847270443124899092917, −2.98530253044838178819832232157, −2.48245986946884601967223136082, −2.01818144563905082777366117290, −1.84120496562006941454486017563, −0.904347861045803115708383857111, −0.53534013729196280025652651376,
0.53534013729196280025652651376, 0.904347861045803115708383857111, 1.84120496562006941454486017563, 2.01818144563905082777366117290, 2.48245986946884601967223136082, 2.98530253044838178819832232157, 3.02755718847270443124899092917, 3.19758381824163121236584394200, 3.46198149452825133429845054148, 3.83947385823145028259015544178, 4.01903251288157073759077382382, 4.48495158731571501877475388209, 4.69475435301017811937614412610, 4.74815458656768011716886465042, 5.30304970252868633953811365137, 5.32981745365429528355255310106, 5.45404758597821470960295222444, 5.79465799480767257051592905319, 6.01430082940369843112638691450, 6.11804106913666178614538117482, 6.11862045358734173184993109672, 6.84961743495114197441092533626, 6.89604458435717238916698407724, 7.03866921314571519093449085626, 7.68497850518753325857033312683
