| L(s) = 1 | − 2-s − 4-s − 7·5-s + 3·8-s − 3·9-s + 7·10-s − 16-s − 11·17-s + 3·18-s + 19-s + 7·20-s + 27·25-s + 9·31-s − 5·32-s + 11·34-s + 3·36-s − 38-s − 21·40-s + 21·45-s − 5·49-s − 27·50-s + 8·59-s + 9·61-s − 9·62-s + 7·64-s − 25·67-s + 11·68-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1/2·4-s − 3.13·5-s + 1.06·8-s − 9-s + 2.21·10-s − 1/4·16-s − 2.66·17-s + 0.707·18-s + 0.229·19-s + 1.56·20-s + 27/5·25-s + 1.61·31-s − 0.883·32-s + 1.88·34-s + 1/2·36-s − 0.162·38-s − 3.32·40-s + 3.13·45-s − 5/7·49-s − 3.81·50-s + 1.04·59-s + 1.15·61-s − 1.14·62-s + 7/8·64-s − 3.05·67-s + 1.33·68-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 987696 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 987696 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 + T + p T^{2} \) |
| 3 | $C_2$ | \( 1 + p T^{2} \) |
| 19 | $C_1$ | \( 1 - T \) |
| good | 5 | $C_2$$\times$$C_2$ | \( ( 1 + 3 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 7 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 11 | $C_2^2$ | \( 1 - 6 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 + 16 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2$$\times$$C_2$ | \( ( 1 + 4 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 23 | $C_2^2$ | \( 1 - 4 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 + 15 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2$$\times$$C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 - 4 T + p T^{2} ) \) |
| 37 | $C_2^2$ | \( 1 + 22 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2^2$ | \( 1 + 17 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 - 31 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 + 8 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 101 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$$\times$$C_2$ | \( ( 1 - 13 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 61 | $C_2$$\times$$C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 + 11 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 71 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 13 T + p T^{2} ) \) |
| 73 | $C_2$$\times$$C_2$ | \( ( 1 - T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 79 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 83 | $C_2^2$ | \( 1 - 20 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 - 103 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 + 34 T^{2} + p^{2} T^{4} \) |
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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
−7.66940097960079654968343320759, −7.52528882656985284840224685193, −7.07576097962381727671007390031, −6.53385733939255667419175852884, −6.11934030024318110818189433734, −5.12892818867964033948783241084, −4.75188746992265904775058613465, −4.34326336593079571239619710955, −4.09178962006457974942008104448, −3.53767065607358261145384427024, −2.96679856343838499555264092530, −2.32469560550463480321015561966, −1.03533847979788543800629084484, 0, 0,
1.03533847979788543800629084484, 2.32469560550463480321015561966, 2.96679856343838499555264092530, 3.53767065607358261145384427024, 4.09178962006457974942008104448, 4.34326336593079571239619710955, 4.75188746992265904775058613465, 5.12892818867964033948783241084, 6.11934030024318110818189433734, 6.53385733939255667419175852884, 7.07576097962381727671007390031, 7.52528882656985284840224685193, 7.66940097960079654968343320759
