| L(s) = 1 | − 4·19-s + 25-s − 2·49-s − 4·61-s + 2·121-s + ⋯ |
| L(s) = 1 | − 4·19-s + 25-s − 2·49-s − 4·61-s + 2·121-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{8} \cdot 5^{4} \cdot 19^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{8} \cdot 5^{4} \cdot 19^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(0.07268033942\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.07268033942\) |
| \(L(1)\) |
|
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 | | \( 1 \) |
| 3 | | \( 1 \) |
| 5 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 19 | $C_1$ | \( ( 1 + T )^{4} \) |
| good | 7 | $C_2$ | \( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \) |
| 11 | $C_2^2$ | \( ( 1 - T^{2} + T^{4} )^{2} \) |
| 13 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 17 | $C_2^2$ | \( ( 1 - T^{2} + T^{4} )^{2} \) |
| 23 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 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 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 41 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{4}( 1 + T )^{4} \) |
| 43 | $C_2$ | \( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( ( 1 - T^{2} + T^{4} )^{2} \) |
| 53 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 59 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{4}( 1 + T )^{4} \) |
| 61 | $C_2$ | \( ( 1 + T + T^{2} )^{4} \) |
| 67 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 71 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{4}( 1 + T )^{4} \) |
| 73 | $C_2$ | \( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \) |
| 79 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{4}( 1 + T )^{4} \) |
| 83 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 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
−6.20747554361759060445944062565, −6.12026566701793130311273261169, −6.08903252918108283469238454761, −5.81229029049140974111447322922, −5.46186477441198323436163220623, −5.24886438150428821568872851706, −4.98500791008912114212561627853, −4.70487530638938024073793241914, −4.61055035300875020600042858018, −4.60067767604199006078697976827, −4.30987823633769286770132987369, −4.13384569196571870009500822415, −3.71842054118447400633883701456, −3.66739662031616528943532530206, −3.30941705912679492089249100270, −3.20915293388545543579034120039, −2.90843591576453859496728046069, −2.54231239514218998236264174133, −2.33998007041311178944670422504, −2.18697013244615416469418341715, −2.01473288901324119161482277346, −1.46942692113537739013303843640, −1.37710403767382326818834437701, −1.10068458968317322435587862502, −0.097022878858990577197837526510,
0.097022878858990577197837526510, 1.10068458968317322435587862502, 1.37710403767382326818834437701, 1.46942692113537739013303843640, 2.01473288901324119161482277346, 2.18697013244615416469418341715, 2.33998007041311178944670422504, 2.54231239514218998236264174133, 2.90843591576453859496728046069, 3.20915293388545543579034120039, 3.30941705912679492089249100270, 3.66739662031616528943532530206, 3.71842054118447400633883701456, 4.13384569196571870009500822415, 4.30987823633769286770132987369, 4.60067767604199006078697976827, 4.61055035300875020600042858018, 4.70487530638938024073793241914, 4.98500791008912114212561627853, 5.24886438150428821568872851706, 5.46186477441198323436163220623, 5.81229029049140974111447322922, 6.08903252918108283469238454761, 6.12026566701793130311273261169, 6.20747554361759060445944062565
