| L(s) = 1 | − 2-s − 2·3-s − 2·5-s + 2·6-s + 7-s + 8-s + 9-s + 2·10-s − 2·11-s + 4·13-s − 14-s + 4·15-s − 16-s + 3·17-s − 18-s − 4·19-s − 2·21-s + 2·22-s − 23-s − 2·24-s − 4·26-s − 2·27-s − 4·30-s − 31-s + 4·33-s − 3·34-s − 2·35-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1.15·3-s − 0.894·5-s + 0.816·6-s + 0.377·7-s + 0.353·8-s + 1/3·9-s + 0.632·10-s − 0.603·11-s + 1.10·13-s − 0.267·14-s + 1.03·15-s − 1/4·16-s + 0.727·17-s − 0.235·18-s − 0.917·19-s − 0.436·21-s + 0.426·22-s − 0.208·23-s − 0.408·24-s − 0.784·26-s − 0.384·27-s − 0.730·30-s − 0.179·31-s + 0.696·33-s − 0.514·34-s − 0.338·35-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 388 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 388 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.1982006890\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1982006890\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 + T + T^{2} \) |
| 97 | $C_1$$\times$$C_2$ | \( ( 1 + T )( 1 - 14 T + p T^{2} ) \) |
| good | 3 | $C_2$$\times$$C_2$ | \( ( 1 - T + p T^{2} )( 1 + p T + p T^{2} ) \) |
| 5 | $D_{4}$ | \( 1 + 2 T + 4 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 7 | $D_{4}$ | \( 1 - T - p T^{3} + p^{2} T^{4} \) |
| 11 | $D_{4}$ | \( 1 + 2 T + T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 13 | $D_{4}$ | \( 1 - 4 T + 12 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 17 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 19 | $D_{4}$ | \( 1 + 4 T + 20 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 + T - 8 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 + 40 T^{2} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + T + 14 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 41 | $C_2$$\times$$C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 53 | $C_2$$\times$$C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) |
| 59 | $D_{4}$ | \( 1 + 6 T - 20 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 37 T^{2} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 14 T + 116 T^{2} - 14 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 6 T + 94 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 8 T + 42 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 83 | $D_{4}$ | \( 1 - 10 T + 124 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 89 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
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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
−19.9804003514, −19.0844364656, −18.7088738959, −18.2748943046, −17.3626593031, −17.2067930848, −16.3942894530, −15.8395495108, −15.3299673196, −14.4018758175, −13.5620147753, −12.7012057581, −12.0306318929, −11.2513554638, −10.8900468160, −10.1528184391, −9.00839466888, −8.16229040744, −7.55108787382, −6.28409327055, −5.34389401817, −4.02372308969,
4.02372308969, 5.34389401817, 6.28409327055, 7.55108787382, 8.16229040744, 9.00839466888, 10.1528184391, 10.8900468160, 11.2513554638, 12.0306318929, 12.7012057581, 13.5620147753, 14.4018758175, 15.3299673196, 15.8395495108, 16.3942894530, 17.2067930848, 17.3626593031, 18.2748943046, 18.7088738959, 19.0844364656, 19.9804003514
