| L(s) = 1 | − 2-s + 3-s + 4-s − 5-s − 6-s − 3·7-s − 3·8-s − 2·9-s + 10-s + 11-s + 12-s + 4·13-s + 3·14-s − 15-s + 16-s − 3·17-s + 2·18-s + 5·19-s − 20-s − 3·21-s − 22-s + 7·23-s − 3·24-s + 25-s − 4·26-s − 2·27-s − 3·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.577·3-s + 1/2·4-s − 0.447·5-s − 0.408·6-s − 1.13·7-s − 1.06·8-s − 2/3·9-s + 0.316·10-s + 0.301·11-s + 0.288·12-s + 1.10·13-s + 0.801·14-s − 0.258·15-s + 1/4·16-s − 0.727·17-s + 0.471·18-s + 1.14·19-s − 0.223·20-s − 0.654·21-s − 0.213·22-s + 1.45·23-s − 0.612·24-s + 1/5·25-s − 0.784·26-s − 0.384·27-s − 0.566·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1311 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1311 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.4297968564\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4297968564\) |
| \(L(\frac{3}{2})\) |
|
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$ | \( ( 1 - T )( 1 + p T^{2} ) \) |
| 19 | $C_1$$\times$$C_2$ | \( ( 1 - T )( 1 - 4 T + p T^{2} ) \) |
| 23 | $C_1$$\times$$C_2$ | \( ( 1 + T )( 1 - 8 T + p T^{2} ) \) |
| good | 2 | $D_{4}$ | \( 1 + T + p T^{3} + p^{2} T^{4} \) |
| 5 | $D_{4}$ | \( 1 + T + p T^{3} + p^{2} T^{4} \) |
| 7 | $D_{4}$ | \( 1 + 3 T + 6 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 11 | $D_{4}$ | \( 1 - T + 18 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 17 | $D_{4}$ | \( 1 + 3 T + 8 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 31 | $D_{4}$ | \( 1 + 4 T - 2 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 41 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 43 | $D_{4}$ | \( 1 + 3 T - 46 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 15 T + 126 T^{2} + 15 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 6 T + 10 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 4 T - 42 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 3 T + 16 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 71 | $D_{4}$ | \( 1 - 18 T + 190 T^{2} - 18 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + T + 100 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2$$\times$$C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
| 83 | $D_{4}$ | \( 1 - 4 T + 38 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 89 | $C_2$$\times$$C_2$ | \( ( 1 - 18 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 97 | $D_{4}$ | \( 1 + 8 T + 46 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
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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
−19.7092208398, −19.0125231070, −18.4378872025, −18.1037422079, −17.4259915387, −16.5998120396, −16.2876048575, −15.7328537857, −15.0416227717, −14.7661898673, −13.7627170061, −13.2516592743, −12.7095385146, −11.6946391199, −11.3876093516, −10.7269635551, −9.55065246533, −9.30018477949, −8.66778524539, −8.03683299628, −6.89567213689, −6.47422703497, −5.39188780808, −3.63259724945, −2.92900322582,
2.92900322582, 3.63259724945, 5.39188780808, 6.47422703497, 6.89567213689, 8.03683299628, 8.66778524539, 9.30018477949, 9.55065246533, 10.7269635551, 11.3876093516, 11.6946391199, 12.7095385146, 13.2516592743, 13.7627170061, 14.7661898673, 15.0416227717, 15.7328537857, 16.2876048575, 16.5998120396, 17.4259915387, 18.1037422079, 18.4378872025, 19.0125231070, 19.7092208398
