| L(s) = 1 | − 2·2-s + 2·3-s + 4·5-s − 4·6-s − 4·7-s + 4·8-s + 9-s − 8·10-s + 2·11-s + 2·13-s + 8·14-s + 8·15-s − 4·16-s + 2·17-s − 2·18-s + 4·19-s − 8·21-s − 4·22-s − 8·23-s + 8·24-s + 3·25-s − 4·26-s + 2·27-s − 6·29-s − 16·30-s + 12·31-s + 4·33-s + ⋯ |
| L(s) = 1 | − 1.41·2-s + 1.15·3-s + 1.78·5-s − 1.63·6-s − 1.51·7-s + 1.41·8-s + 1/3·9-s − 2.52·10-s + 0.603·11-s + 0.554·13-s + 2.13·14-s + 2.06·15-s − 16-s + 0.485·17-s − 0.471·18-s + 0.917·19-s − 1.74·21-s − 0.852·22-s − 1.66·23-s + 1.63·24-s + 3/5·25-s − 0.784·26-s + 0.384·27-s − 1.11·29-s − 2.92·30-s + 2.15·31-s + 0.696·33-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5687 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5687 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.6040601613\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6040601613\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 11 | $C_1$ | \( ( 1 - T )^{2} \) |
| 47 | $C_1$$\times$$C_2$ | \( ( 1 + T )( 1 - 8 T + p T^{2} ) \) |
| good | 2 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + p T + p T^{2} ) \) |
| 3 | $C_2$$\times$$C_2$ | \( ( 1 - p T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 5 | $C_2$$\times$$C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 - T + p T^{2} ) \) |
| 7 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 13 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 17 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 19 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + p T^{2} ) \) |
| 23 | $C_2$$\times$$C_2$ | \( ( 1 + T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 29 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 31 | $C_2$$\times$$C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 - 5 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 41 | $C_2$$\times$$C_2$ | \( ( 1 + 6 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 - 5 T + p T^{2} )^{2} \) |
| 61 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 - 15 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 71 | $C_2$$\times$$C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 73 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + p T^{2} ) \) |
| 79 | $C_2$$\times$$C_2$ | \( ( 1 + 10 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 89 | $C_2$$\times$$C_2$ | \( ( 1 - 15 T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 - T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
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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
−17.3578238615, −17.0336103204, −16.5873541880, −15.9140726033, −15.5197416819, −14.4805292129, −14.1792914635, −13.5686390571, −13.5110346821, −13.0204378500, −12.1094148704, −11.4512586103, −10.0560138700, −10.0355090972, −9.76146545711, −9.37585903041, −8.60353961929, −8.35425554610, −7.58939326119, −6.36261389471, −6.26841229097, −5.16274252995, −3.81906067500, −2.97738750512, −1.69080561744,
1.69080561744, 2.97738750512, 3.81906067500, 5.16274252995, 6.26841229097, 6.36261389471, 7.58939326119, 8.35425554610, 8.60353961929, 9.37585903041, 9.76146545711, 10.0355090972, 10.0560138700, 11.4512586103, 12.1094148704, 13.0204378500, 13.5110346821, 13.5686390571, 14.1792914635, 14.4805292129, 15.5197416819, 15.9140726033, 16.5873541880, 17.0336103204, 17.3578238615
