| L(s) = 1 | − 2-s − 2·3-s − 4-s + 5-s + 2·6-s − 3·7-s + 8-s + 2·9-s − 10-s + 4·11-s + 2·12-s + 13-s + 3·14-s − 2·15-s − 16-s − 2·18-s − 19-s − 20-s + 6·21-s − 4·22-s − 2·24-s + 25-s − 26-s − 6·27-s + 3·28-s + 2·29-s + 2·30-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1.15·3-s − 1/2·4-s + 0.447·5-s + 0.816·6-s − 1.13·7-s + 0.353·8-s + 2/3·9-s − 0.316·10-s + 1.20·11-s + 0.577·12-s + 0.277·13-s + 0.801·14-s − 0.516·15-s − 1/4·16-s − 0.471·18-s − 0.229·19-s − 0.223·20-s + 1.30·21-s − 0.852·22-s − 0.408·24-s + 1/5·25-s − 0.196·26-s − 1.15·27-s + 0.566·28-s + 0.371·29-s + 0.365·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 389 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 389 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.1979862013\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1979862013\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 389 | $C_1$$\times$$C_2$ | \( ( 1 + T )( 1 + 10 T + p T^{2} ) \) |
| good | 2 | $C_2$$\times$$C_2$ | \( ( 1 - T + p T^{2} )( 1 + p T + p T^{2} ) \) |
| 3 | $C_2^2$ | \( 1 + 2 T + 2 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 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 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 13 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 17 | $C_2^2$ | \( 1 + 10 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 23 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 31 | $D_{4}$ | \( 1 + 4 T + 10 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2$$\times$$C_2$ | \( ( 1 + 2 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 41 | $D_{4}$ | \( 1 - 5 T + 28 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - 4 T + 6 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 53 | $C_2$$\times$$C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 59 | $D_{4}$ | \( 1 + 3 T - 22 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 67 | $D_{4}$ | \( 1 + T + 42 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 6 T - 10 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 5 T + 60 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 9 T + 118 T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 12 T + 82 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 89 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 18 T + p T^{2} ) \) |
| 97 | $D_{4}$ | \( 1 + T + 112 T^{2} + p T^{3} + p^{2} T^{4} \) |
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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
−19.6170191729, −19.3498183071, −18.6235408130, −18.1322059841, −17.4045507751, −17.1824093561, −16.5664957431, −15.9272149391, −15.2180879131, −14.1516609715, −13.6128208412, −12.7753631653, −12.2044917017, −11.4012010263, −10.6121010392, −9.77317652912, −9.28854322905, −8.58060022192, −7.05058226774, −6.34866827136, −5.47801220030, −3.96910734375,
3.96910734375, 5.47801220030, 6.34866827136, 7.05058226774, 8.58060022192, 9.28854322905, 9.77317652912, 10.6121010392, 11.4012010263, 12.2044917017, 12.7753631653, 13.6128208412, 14.1516609715, 15.2180879131, 15.9272149391, 16.5664957431, 17.1824093561, 17.4045507751, 18.1322059841, 18.6235408130, 19.3498183071, 19.6170191729
