| L(s) = 1 | + 4·7-s + 4·13-s − 16-s − 4·37-s + 8·49-s + 16·91-s − 4·103-s − 4·112-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 8·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s − 4·208-s + 211-s + ⋯ |
| L(s) = 1 | + 4·7-s + 4·13-s − 16-s − 4·37-s + 8·49-s + 16·91-s − 4·103-s − 4·112-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 8·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s − 4·208-s + 211-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 3^{8} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 3^{8} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(2.854206125\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.854206125\) |
| \(L(1)\) |
|
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 | $C_2^2$ | \( 1 + T^{4} \) |
| 3 | | \( 1 \) |
| 5 | | \( 1 \) |
| good | 7 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{4}( 1 + T^{2} )^{2} \) |
| 11 | $C_2^2$ | \( ( 1 + T^{4} )^{2} \) |
| 13 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{4}( 1 + T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( ( 1 + T^{4} )^{2} \) |
| 19 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{4}( 1 + T )^{4} \) |
| 23 | $C_2^2$ | \( ( 1 + T^{4} )^{2} \) |
| 29 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{4}( 1 + T )^{4} \) |
| 31 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{4}( 1 + T )^{4} \) |
| 37 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{4}( 1 + T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( ( 1 + T^{4} )^{2} \) |
| 43 | $C_2^2$ | \( ( 1 + T^{4} )^{2} \) |
| 47 | $C_2^2$ | \( ( 1 + T^{4} )^{2} \) |
| 53 | $C_2^2$ | \( ( 1 + T^{4} )^{2} \) |
| 59 | $C_2^2$ | \( ( 1 + T^{4} )^{2} \) |
| 61 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{4}( 1 + T )^{4} \) |
| 67 | $C_2^2$ | \( ( 1 + T^{4} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 73 | $C_2^2$ | \( ( 1 + T^{4} )^{2} \) |
| 79 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 83 | $C_2^2$ | \( ( 1 + T^{4} )^{2} \) |
| 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
−6.82163482035063539648652382103, −6.72604773295394192365750713765, −6.31269381469828824786763980612, −6.13288875337869690260851175448, −5.81681196100221932068276425534, −5.69006235946435152858730382070, −5.62105468480300655623066643085, −5.20080844306601801970805807940, −5.03585486811106907809975034803, −4.99140682932471197329706964631, −4.56295965746085436971958449694, −4.47218056618065224555619094941, −4.30613868598206531911541445269, −3.97170366370086445782045915606, −3.63321752741537757675417626531, −3.55958993731141655793156347218, −3.41868319869982043926084036739, −3.05662780229486034017187702195, −2.35199168979366122086570932528, −2.32188695783414856389982357589, −2.04303132672494496543575639120, −1.52902223660804451088843312210, −1.44817319801647164540447848030, −1.36612398721585516636666245689, −1.07594996192474696148767366100,
1.07594996192474696148767366100, 1.36612398721585516636666245689, 1.44817319801647164540447848030, 1.52902223660804451088843312210, 2.04303132672494496543575639120, 2.32188695783414856389982357589, 2.35199168979366122086570932528, 3.05662780229486034017187702195, 3.41868319869982043926084036739, 3.55958993731141655793156347218, 3.63321752741537757675417626531, 3.97170366370086445782045915606, 4.30613868598206531911541445269, 4.47218056618065224555619094941, 4.56295965746085436971958449694, 4.99140682932471197329706964631, 5.03585486811106907809975034803, 5.20080844306601801970805807940, 5.62105468480300655623066643085, 5.69006235946435152858730382070, 5.81681196100221932068276425534, 6.13288875337869690260851175448, 6.31269381469828824786763980612, 6.72604773295394192365750713765, 6.82163482035063539648652382103
