| L(s) = 1 | − 3-s − 4-s − 5-s + 2·8-s − 2·9-s − 4·11-s + 12-s + 15-s + 16-s − 4·17-s + 8·19-s + 20-s + 8·23-s − 2·24-s − 2·25-s + 2·27-s − 31-s − 4·32-s + 4·33-s + 2·36-s − 2·40-s + 4·41-s + 12·43-s + 4·44-s + 2·45-s − 48-s − 14·49-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 1/2·4-s − 0.447·5-s + 0.707·8-s − 2/3·9-s − 1.20·11-s + 0.288·12-s + 0.258·15-s + 1/4·16-s − 0.970·17-s + 1.83·19-s + 0.223·20-s + 1.66·23-s − 0.408·24-s − 2/5·25-s + 0.384·27-s − 0.179·31-s − 0.707·32-s + 0.696·33-s + 1/3·36-s − 0.316·40-s + 0.624·41-s + 1.82·43-s + 0.603·44-s + 0.298·45-s − 0.144·48-s − 2·49-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.3882263855\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3882263855\) |
| \(L(\frac{3}{2})\) |
|
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$ | \( ( 1 + T )( 1 + p T^{2} ) \) |
| 5 | $C_1$$\times$$C_2$ | \( ( 1 - T )( 1 + 2 T + p T^{2} ) \) |
| 31 | $C_1$$\times$$C_2$ | \( ( 1 + T )( 1 + p T^{2} ) \) |
| good | 7 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 11 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 17 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 23 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + p T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 41 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 43 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 - 4 T + p T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 53 | $C_2$$\times$$C_2$ | \( ( 1 + 6 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 59 | $C_2$$\times$$C_2$ | \( ( 1 + 4 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 61 | $C_2$$\times$$C_2$ | \( ( 1 + 2 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 73 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 79 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 83 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 - 8 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| show more | | |
| show less | | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−19.4991076981, −19.1104624590, −18.1277233219, −18.1058537571, −17.2359534775, −16.9117681972, −15.9590260302, −15.8160898872, −15.0569175078, −14.0069258505, −13.9098647616, −12.8826816262, −12.6461787614, −11.5293542519, −11.0627456915, −10.6789224512, −9.52345167581, −9.05649826746, −7.76857292338, −7.66488013442, −6.32858579421, −5.23920392625, −4.74203912194, −3.16188554299,
3.16188554299, 4.74203912194, 5.23920392625, 6.32858579421, 7.66488013442, 7.76857292338, 9.05649826746, 9.52345167581, 10.6789224512, 11.0627456915, 11.5293542519, 12.6461787614, 12.8826816262, 13.9098647616, 14.0069258505, 15.0569175078, 15.8160898872, 15.9590260302, 16.9117681972, 17.2359534775, 18.1058537571, 18.1277233219, 19.1104624590, 19.4991076981
