| L(s) = 1 | + 3-s + 3·13-s + 2·19-s − 25-s − 27-s − 2·31-s + 3·39-s − 3·43-s − 49-s + 2·57-s − 61-s + 67-s − 73-s − 75-s − 79-s − 81-s − 2·93-s + 2·103-s − 2·121-s + 127-s − 3·129-s + 131-s + 137-s + 139-s − 147-s + 149-s + 151-s + ⋯ |
| L(s) = 1 | + 3-s + 3·13-s + 2·19-s − 25-s − 27-s − 2·31-s + 3·39-s − 3·43-s − 49-s + 2·57-s − 61-s + 67-s − 73-s − 75-s − 79-s − 81-s − 2·93-s + 2·103-s − 2·121-s + 127-s − 3·129-s + 131-s + 137-s + 139-s − 147-s + 149-s + 151-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 831744 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 831744 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(1.453418167\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.453418167\) |
| \(L(1)\) |
|
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 | $C_2$ | \( 1 - T + T^{2} \) |
| 19 | $C_1$ | \( ( 1 - T )^{2} \) |
| good | 5 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 7 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 11 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 13 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 - T + T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 23 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 29 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 31 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 41 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 43 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{2}( 1 + T + T^{2} ) \) |
| 47 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 53 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 59 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 61 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{2}( 1 - T + T^{2} ) \) |
| 67 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 + T + T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 73 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{2}( 1 - T + T^{2} ) \) |
| 79 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{2}( 1 - T + T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 97 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.36352088995836544751663582398, −10.08354571323028296743835279195, −9.519992304093079476585738558546, −9.147564389348741810055681283591, −8.861772297385606034400995761666, −8.440459865224329581018570531611, −8.022366461424400834894634235743, −7.78596569086055731794282189680, −7.17143060953787714839565598715, −6.74209416609971380991848882778, −5.99165484588640809411999551258, −5.94050629678203828923026731400, −5.31315066045061950650358004513, −4.82640766478715929900894341162, −3.84920861665654669624569662995, −3.53606581861919412649860763948, −3.45640035068502334256842326387, −2.73948225266767511004073887190, −1.57213550181728894720236107347, −1.55534855442202624401662324813,
1.55534855442202624401662324813, 1.57213550181728894720236107347, 2.73948225266767511004073887190, 3.45640035068502334256842326387, 3.53606581861919412649860763948, 3.84920861665654669624569662995, 4.82640766478715929900894341162, 5.31315066045061950650358004513, 5.94050629678203828923026731400, 5.99165484588640809411999551258, 6.74209416609971380991848882778, 7.17143060953787714839565598715, 7.78596569086055731794282189680, 8.022366461424400834894634235743, 8.440459865224329581018570531611, 8.861772297385606034400995761666, 9.147564389348741810055681283591, 9.519992304093079476585738558546, 10.08354571323028296743835279195, 10.36352088995836544751663582398
