| L(s) = 1 | − 2-s + 2·3-s + 5-s − 2·6-s + 7-s + 8-s + 3·9-s − 10-s − 2·11-s − 14-s + 2·15-s − 16-s − 3·18-s − 2·19-s + 2·21-s + 2·22-s + 2·24-s + 25-s + 4·27-s − 2·30-s − 4·33-s + 35-s + 4·37-s + 2·38-s + 40-s − 41-s − 2·42-s + ⋯ |
| L(s) = 1 | − 2-s + 2·3-s + 5-s − 2·6-s + 7-s + 8-s + 3·9-s − 10-s − 2·11-s − 14-s + 2·15-s − 16-s − 3·18-s − 2·19-s + 2·21-s + 2·22-s + 2·24-s + 25-s + 4·27-s − 2·30-s − 4·33-s + 35-s + 4·37-s + 2·38-s + 40-s − 41-s − 2·42-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2178576 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2178576 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(1.611497051\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.611497051\) |
| \(L(1)\) |
|
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 | $C_2$ | \( 1 + T + T^{2} \) |
| 3 | $C_1$ | \( ( 1 - T )^{2} \) |
| 41 | $C_2$ | \( 1 + T + T^{2} \) |
| good | 5 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 + T + T^{2} ) \) |
| 7 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 + T + T^{2} ) \) |
| 11 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 17 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 19 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 37 | $C_1$ | \( ( 1 - T )^{4} \) |
| 43 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 47 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 + T + T^{2} ) \) |
| 53 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 59 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 79 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 + T + T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 89 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 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
−9.852093425342553834941792903900, −9.268644910146842581304808520052, −9.028596381768058299065497964343, −8.800073083613172759744438001728, −8.252923905727239576752200143767, −8.061525599745425706260764152946, −7.72634811054817307348826255651, −7.41260571836202307387823896206, −7.05209590870693509116555241202, −6.13994456167848882732450426388, −6.06046710155956419848652611576, −5.16020578303014785776161859702, −4.71743425635488370197878838227, −4.27912388403059770962778104930, −4.22281957197685316028017082396, −3.04747995251353979527698997043, −2.59886832209306016413304130508, −2.40623682513382176772668712794, −1.74905797969713757496601520973, −1.26291410200319182373913809526,
1.26291410200319182373913809526, 1.74905797969713757496601520973, 2.40623682513382176772668712794, 2.59886832209306016413304130508, 3.04747995251353979527698997043, 4.22281957197685316028017082396, 4.27912388403059770962778104930, 4.71743425635488370197878838227, 5.16020578303014785776161859702, 6.06046710155956419848652611576, 6.13994456167848882732450426388, 7.05209590870693509116555241202, 7.41260571836202307387823896206, 7.72634811054817307348826255651, 8.061525599745425706260764152946, 8.252923905727239576752200143767, 8.800073083613172759744438001728, 9.028596381768058299065497964343, 9.268644910146842581304808520052, 9.852093425342553834941792903900
