| L(s) = 1 | − 3-s − 4-s − 3·7-s + 2·8-s − 2·11-s + 12-s − 2·13-s + 16-s + 4·17-s − 4·19-s + 3·21-s + 23-s − 2·24-s + 2·25-s + 4·27-s + 3·28-s + 12·29-s − 4·32-s + 2·33-s − 6·37-s + 2·39-s − 8·41-s − 6·43-s + 2·44-s + 4·47-s − 48-s + 6·49-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 1/2·4-s − 1.13·7-s + 0.707·8-s − 0.603·11-s + 0.288·12-s − 0.554·13-s + 1/4·16-s + 0.970·17-s − 0.917·19-s + 0.654·21-s + 0.208·23-s − 0.408·24-s + 2/5·25-s + 0.769·27-s + 0.566·28-s + 2.22·29-s − 0.707·32-s + 0.348·33-s − 0.986·37-s + 0.320·39-s − 1.24·41-s − 0.914·43-s + 0.301·44-s + 0.583·47-s − 0.144·48-s + 6/7·49-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 966 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 966 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.3969487833\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3969487833\) |
| \(L(\frac{3}{2})\) |
|
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$$\times$$C_2$ | \( ( 1 - T )( 1 + T + p T^{2} ) \) |
| 3 | $C_1$$\times$$C_2$ | \( ( 1 - T )( 1 + 2 T + p T^{2} ) \) |
| 7 | $C_1$$\times$$C_2$ | \( ( 1 - T )( 1 + 4 T + p T^{2} ) \) |
| 23 | $C_1$$\times$$C_2$ | \( ( 1 - T )( 1 + p T^{2} ) \) |
| good | 5 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 13 | $D_{4}$ | \( 1 + 2 T - 6 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 17 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 19 | $D_{4}$ | \( 1 + 4 T + 14 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 37 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 41 | $D_{4}$ | \( 1 + 8 T + 46 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 47 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 53 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 59 | $C_2$$\times$$C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 61 | $C_2^2$ | \( 1 - 26 T^{2} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 2 T + 62 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2^2$ | \( 1 - 50 T^{2} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 20 T + 198 T^{2} + 20 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2$$\times$$C_2$ | \( ( 1 - 16 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 89 | $C_2^2$ | \( 1 - 50 T^{2} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 12 T + 118 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
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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
−19.9566954785, −19.3658017746, −18.9673994757, −18.5387216250, −17.6104977510, −17.3535496550, −16.5514626755, −16.3189395498, −15.6386670813, −14.8989727954, −14.1597098421, −13.6086719916, −12.9557220643, −12.3662947651, −11.9665914829, −10.7804508976, −10.2452501109, −9.93594280145, −8.81181327685, −8.16788311990, −7.09377461697, −6.44412641605, −5.34338417472, −4.58611189713, −3.13389750393,
3.13389750393, 4.58611189713, 5.34338417472, 6.44412641605, 7.09377461697, 8.16788311990, 8.81181327685, 9.93594280145, 10.2452501109, 10.7804508976, 11.9665914829, 12.3662947651, 12.9557220643, 13.6086719916, 14.1597098421, 14.8989727954, 15.6386670813, 16.3189395498, 16.5514626755, 17.3535496550, 17.6104977510, 18.5387216250, 18.9673994757, 19.3658017746, 19.9566954785
