| L(s) = 1 | − 2-s − 3-s + 4-s − 2·5-s + 6-s − 4·7-s − 8-s + 2·10-s − 4·11-s − 12-s + 2·13-s + 4·14-s + 2·15-s + 16-s − 4·17-s − 2·19-s − 2·20-s + 4·21-s + 4·22-s − 23-s + 24-s − 2·26-s + 27-s − 4·28-s + 2·29-s − 2·30-s − 2·31-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s + 1/2·4-s − 0.894·5-s + 0.408·6-s − 1.51·7-s − 0.353·8-s + 0.632·10-s − 1.20·11-s − 0.288·12-s + 0.554·13-s + 1.06·14-s + 0.516·15-s + 1/4·16-s − 0.970·17-s − 0.458·19-s − 0.447·20-s + 0.872·21-s + 0.852·22-s − 0.208·23-s + 0.204·24-s − 0.392·26-s + 0.192·27-s − 0.755·28-s + 0.371·29-s − 0.365·30-s − 0.359·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6336 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6336 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | $C_1$ | \( 1 + T \) |
| 3 | $C_2$ | \( 1 + T + T^{2} \) |
| 11 | $C_1$$\times$$C_2$ | \( ( 1 + T )( 1 + 3 T + p T^{2} ) \) |
| good | 5 | $D_{4}$ | \( 1 + 2 T + 4 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 7 | $D_{4}$ | \( 1 + 4 T + 11 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2$$\times$$C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 17 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 19 | $D_{4}$ | \( 1 + 2 T + 18 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 + T - 32 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 - 2 T + 13 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 2 T - 24 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 - 3 T - 2 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 8 T + 46 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - 4 T - 12 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 5 T + 82 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 6 T - 2 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 14 T^{2} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + T - 40 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 2 T + 81 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 73 | $D_{4}$ | \( 1 - 3 T + 112 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 8 T + 51 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - T + 64 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 20 T + 223 T^{2} + 20 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 4 T - 42 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
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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
−17.3255000072, −16.9441916616, −16.3317567504, −15.9439451971, −15.6507279975, −15.3193763839, −14.5786064046, −13.6323934065, −13.1893512327, −12.7948932388, −12.1184508968, −11.7158879729, −10.8507119640, −10.7594641202, −9.99630618545, −9.47549642105, −8.63730854446, −8.28747136881, −7.40188821007, −6.89552073034, −6.20040224745, −5.61176083805, −4.47740975400, −3.55456495973, −2.59130542311, 0,
2.59130542311, 3.55456495973, 4.47740975400, 5.61176083805, 6.20040224745, 6.89552073034, 7.40188821007, 8.28747136881, 8.63730854446, 9.47549642105, 9.99630618545, 10.7594641202, 10.8507119640, 11.7158879729, 12.1184508968, 12.7948932388, 13.1893512327, 13.6323934065, 14.5786064046, 15.3193763839, 15.6507279975, 15.9439451971, 16.3317567504, 16.9441916616, 17.3255000072
