| L(s) = 1 | − 2-s − 3·3-s + 4-s − 2·5-s + 3·6-s − 2·7-s + 8-s + 4·9-s + 2·10-s − 8·11-s − 3·12-s + 13-s + 2·14-s + 6·15-s − 3·16-s − 17-s − 4·18-s − 3·19-s − 2·20-s + 6·21-s + 8·22-s − 5·23-s − 3·24-s + 4·25-s − 26-s − 2·28-s + 7·29-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1.73·3-s + 1/2·4-s − 0.894·5-s + 1.22·6-s − 0.755·7-s + 0.353·8-s + 4/3·9-s + 0.632·10-s − 2.41·11-s − 0.866·12-s + 0.277·13-s + 0.534·14-s + 1.54·15-s − 3/4·16-s − 0.242·17-s − 0.942·18-s − 0.688·19-s − 0.447·20-s + 1.30·21-s + 1.70·22-s − 1.04·23-s − 0.612·24-s + 4/5·25-s − 0.196·26-s − 0.377·28-s + 1.29·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3138 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3138 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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_1$$\times$$C_2$ | \( ( 1 - T )( 1 + p T + p T^{2} ) \) |
| 3 | $C_1$$\times$$C_2$ | \( ( 1 + T )( 1 + 2 T + p T^{2} ) \) |
| 523 | $C_1$$\times$$C_2$ | \( ( 1 + T )( 1 - 4 T + p T^{2} ) \) |
| good | 5 | $D_{4}$ | \( 1 + 2 T + 2 p T^{3} + p^{2} T^{4} \) |
| 7 | $D_{4}$ | \( 1 + 2 T - 2 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2$$\times$$C_2$ | \( ( 1 + 2 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 13 | $D_{4}$ | \( 1 - T - p T^{3} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 + T + p T^{3} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 + 3 T + 2 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 + 5 T + 26 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 - 7 T + 44 T^{2} - 7 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 37 | $D_{4}$ | \( 1 + 2 T + 42 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + T - 16 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 5 T + 82 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 8 T + 44 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 13 T + 108 T^{2} + 13 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 12 T + 134 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 4 T - 36 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 4 T + 42 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 13 T + 166 T^{2} - 13 p T^{3} + p^{2} T^{4} \) |
| 73 | $C_2$$\times$$C_2$ | \( ( 1 + 6 T + p T^{2} )( 1 + 11 T + p T^{2} ) \) |
| 79 | $D_{4}$ | \( 1 - 8 T + 102 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 2 T + 110 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 89 | $C_4$ | \( 1 + 12 T + 86 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 + 4 T + p T^{2} )( 1 + 18 T + p T^{2} ) \) |
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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
−18.3931241071, −17.8785667400, −17.4420347369, −16.8374150248, −16.2721093174, −16.0469687070, −15.5171724216, −15.3698826741, −14.0127997130, −13.4599326017, −12.7791006484, −12.2999349308, −11.8592994779, −10.9640108847, −10.7830268260, −10.3519997002, −9.74924581348, −8.47882318253, −8.09625462297, −7.32753724656, −6.60994379326, −5.97575442625, −5.07428350226, −4.41333089940, −2.78975465914, 0,
2.78975465914, 4.41333089940, 5.07428350226, 5.97575442625, 6.60994379326, 7.32753724656, 8.09625462297, 8.47882318253, 9.74924581348, 10.3519997002, 10.7830268260, 10.9640108847, 11.8592994779, 12.2999349308, 12.7791006484, 13.4599326017, 14.0127997130, 15.3698826741, 15.5171724216, 16.0469687070, 16.2721093174, 16.8374150248, 17.4420347369, 17.8785667400, 18.3931241071
