L(s) = 1 | − 2-s − 2·3-s + 4-s + 2·5-s + 2·6-s + 3·7-s − 3·8-s − 2·9-s − 2·10-s − 2·12-s + 2·13-s − 3·14-s − 4·15-s + 16-s − 7·17-s + 2·18-s − 4·19-s + 2·20-s − 6·21-s + 3·23-s + 6·24-s + 2·25-s − 2·26-s + 10·27-s + 3·28-s − 2·29-s + 4·30-s + ⋯ |
L(s) = 1 | − 0.707·2-s − 1.15·3-s + 1/2·4-s + 0.894·5-s + 0.816·6-s + 1.13·7-s − 1.06·8-s − 2/3·9-s − 0.632·10-s − 0.577·12-s + 0.554·13-s − 0.801·14-s − 1.03·15-s + 1/4·16-s − 1.69·17-s + 0.471·18-s − 0.917·19-s + 0.447·20-s − 1.30·21-s + 0.625·23-s + 1.22·24-s + 2/5·25-s − 0.392·26-s + 1.92·27-s + 0.566·28-s − 0.371·29-s + 0.730·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1123 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1123 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.3358100703\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.3358100703\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
---|
bad | 1123 | $C_1$$\times$$C_2$ | \( ( 1 + T )( 1 - 52 T + p T^{2} ) \) |
good | 2 | $D_{4}$ | \( 1 + T + p T^{3} + p^{2} T^{4} \) |
| 3 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 5 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 7 | $D_{4}$ | \( 1 - 3 T + 6 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 17 | $C_2^2$ | \( 1 + 7 T + 32 T^{2} + 7 p T^{3} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 + 4 T + 22 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 - 3 T + 22 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 31 | $D_{4}$ | \( 1 - 4 T + 54 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 41 | $D_{4}$ | \( 1 - T + 24 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + p T^{2} ) \) |
| 47 | $D_{4}$ | \( 1 - 4 T + 6 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 4 T + 62 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 3 T - 14 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 67 | $D_{4}$ | \( 1 - 7 T + 130 T^{2} - 7 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 6 T + 22 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 2 T - 102 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 3 T - 18 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2^2$ | \( 1 + 110 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 + 14 T^{2} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 8 T + 78 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
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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
−19.8589322797, −19.0738220745, −18.3927408852, −17.8836523525, −17.5618295278, −17.2572835138, −16.8616221926, −15.9594610685, −15.3950883421, −14.7716799364, −14.0944726136, −13.4770627391, −12.6626952282, −11.8108724426, −11.4026417065, −10.8477425448, −10.5144035914, −9.16796495712, −8.87340843902, −8.19476477713, −6.84038187370, −6.21447203441, −5.61477356340, −4.62775681960, −2.41842817209,
2.41842817209, 4.62775681960, 5.61477356340, 6.21447203441, 6.84038187370, 8.19476477713, 8.87340843902, 9.16796495712, 10.5144035914, 10.8477425448, 11.4026417065, 11.8108724426, 12.6626952282, 13.4770627391, 14.0944726136, 14.7716799364, 15.3950883421, 15.9594610685, 16.8616221926, 17.2572835138, 17.5618295278, 17.8836523525, 18.3927408852, 19.0738220745, 19.8589322797