| L(s) = 1 | − 2·4-s + 3·5-s + 7-s − 2·9-s + 11-s + 3·13-s − 3·19-s − 6·20-s − 23-s + 25-s − 3·27-s − 2·28-s − 9·29-s + 2·31-s + 3·35-s + 4·36-s − 2·37-s + 9·41-s − 3·43-s − 2·44-s − 6·45-s − 47-s − 5·49-s − 6·52-s + 6·53-s + 3·55-s − 16·59-s + ⋯ |
| L(s) = 1 | − 4-s + 1.34·5-s + 0.377·7-s − 2/3·9-s + 0.301·11-s + 0.832·13-s − 0.688·19-s − 1.34·20-s − 0.208·23-s + 1/5·25-s − 0.577·27-s − 0.377·28-s − 1.67·29-s + 0.359·31-s + 0.507·35-s + 2/3·36-s − 0.328·37-s + 1.40·41-s − 0.457·43-s − 0.301·44-s − 0.894·45-s − 0.145·47-s − 5/7·49-s − 0.832·52-s + 0.824·53-s + 0.404·55-s − 2.08·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 417747 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 417747 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | $C_1$$\times$$C_2$ | \( ( 1 + T )( 1 - T + p T^{2} ) \) |
| 11 | $C_1$$\times$$C_2$ | \( ( 1 + T )( 1 - 2 T + p T^{2} ) \) |
| 12659 | $C_1$$\times$$C_2$ | \( ( 1 - T )( 1 + 78 T + p T^{2} ) \) |
| good | 2 | $C_2^2$ | \( 1 + p T^{2} + p^{2} T^{4} \) |
| 5 | $C_2^2$ | \( 1 - 3 T + 8 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 7 | $D_{4}$ | \( 1 - T + 6 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 13 | $D_{4}$ | \( 1 - 3 T + 2 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 19 | $D_{4}$ | \( 1 + 3 T + 7 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 + T - 22 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 9 T + p T^{2} ) \) |
| 31 | $D_{4}$ | \( 1 - 2 T + 10 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 2 T + 48 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_4$ | \( 1 - 9 T + 71 T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 3 T - T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + T + 3 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 6 T + 10 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 16 T + 142 T^{2} + 16 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 5 T - 23 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 13 T + p T^{2} ) \) |
| 71 | $D_{4}$ | \( 1 - 12 T + 160 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 73 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 79 | $D_{4}$ | \( 1 + 17 T + 183 T^{2} + 17 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 10 T + 118 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 + 126 T^{2} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 19 T + 260 T^{2} - 19 p T^{3} + p^{2} T^{4} \) |
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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
−13.2094431603, −12.8143529423, −12.2167274905, −11.7856798511, −11.2717885061, −10.8692538775, −10.6908072143, −9.90270303313, −9.62718025697, −9.24676157287, −8.99236463871, −8.49387807785, −8.03838298138, −7.66602157266, −6.87508605557, −6.41986268262, −5.90157600933, −5.61727301737, −5.21654463468, −4.45374970539, −4.09823499364, −3.50102775246, −2.68027368201, −1.97724961328, −1.41324581821, 0,
1.41324581821, 1.97724961328, 2.68027368201, 3.50102775246, 4.09823499364, 4.45374970539, 5.21654463468, 5.61727301737, 5.90157600933, 6.41986268262, 6.87508605557, 7.66602157266, 8.03838298138, 8.49387807785, 8.99236463871, 9.24676157287, 9.62718025697, 9.90270303313, 10.6908072143, 10.8692538775, 11.2717885061, 11.7856798511, 12.2167274905, 12.8143529423, 13.2094431603
