| L(s) = 1 | − 2·4-s + 4·16-s − 8·19-s + 12·23-s + 12·29-s − 16·43-s − 4·49-s − 24·53-s − 8·64-s − 16·67-s − 28·73-s + 16·76-s − 24·92-s + 20·97-s + 12·101-s − 24·116-s + 20·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 8·169-s + ⋯ |
| L(s) = 1 | − 4-s + 16-s − 1.83·19-s + 2.50·23-s + 2.22·29-s − 2.43·43-s − 4/7·49-s − 3.29·53-s − 64-s − 1.95·67-s − 3.27·73-s + 1.83·76-s − 2.50·92-s + 2.03·97-s + 1.19·101-s − 2.22·116-s + 1.81·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 8/13·169-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3240000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3240000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.057095402\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.057095402\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 + p T^{2} \) |
| 3 | | \( 1 \) |
| 5 | | \( 1 \) |
| good | 7 | $C_2^2$ | \( 1 + 4 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 20 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 - 8 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - 26 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 + 10 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 - 56 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2^2$ | \( 1 + 16 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 - 116 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 50 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 + 14 T + p T^{2} )^{2} \) |
| 79 | $C_2^2$ | \( 1 - 86 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 - 18 T + p T^{2} )( 1 + 18 T + p T^{2} ) \) |
| 89 | $C_2^2$ | \( 1 - 128 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
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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
−9.386751054799445554481821199265, −8.844598893443983252517936360287, −8.842740747636572936086464735644, −8.297745361931099178344493263400, −8.205361081314304704624939307082, −7.32720518326110921106398053996, −7.32269319567083315819448210421, −6.47535308940094793654442311750, −6.34248834739458434702500006255, −6.03114680424289339977062113057, −5.14775454467630311450647549242, −4.88939408086467286081256163084, −4.60229434698942712152672181359, −4.33785091652723087491702960495, −3.49109620217490621809330320585, −2.96411411523641432886475456508, −2.94560662887635769265180964005, −1.76223914003643151288611390395, −1.35850624110957309026676314902, −0.41031757181414092715319148318,
0.41031757181414092715319148318, 1.35850624110957309026676314902, 1.76223914003643151288611390395, 2.94560662887635769265180964005, 2.96411411523641432886475456508, 3.49109620217490621809330320585, 4.33785091652723087491702960495, 4.60229434698942712152672181359, 4.88939408086467286081256163084, 5.14775454467630311450647549242, 6.03114680424289339977062113057, 6.34248834739458434702500006255, 6.47535308940094793654442311750, 7.32269319567083315819448210421, 7.32720518326110921106398053996, 8.205361081314304704624939307082, 8.297745361931099178344493263400, 8.842740747636572936086464735644, 8.844598893443983252517936360287, 9.386751054799445554481821199265
