| L(s) = 1 | − 3-s − 4-s − 5-s − 3·7-s + 2·8-s − 9-s − 11-s + 12-s + 4·13-s + 15-s + 16-s + 3·17-s − 6·19-s + 20-s + 3·21-s + 4·23-s − 2·24-s − 2·25-s + 3·28-s + 3·29-s + 31-s − 4·32-s + 33-s + 3·35-s + 36-s + 3·37-s − 4·39-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 1/2·4-s − 0.447·5-s − 1.13·7-s + 0.707·8-s − 1/3·9-s − 0.301·11-s + 0.288·12-s + 1.10·13-s + 0.258·15-s + 1/4·16-s + 0.727·17-s − 1.37·19-s + 0.223·20-s + 0.654·21-s + 0.834·23-s − 0.408·24-s − 2/5·25-s + 0.566·28-s + 0.557·29-s + 0.179·31-s − 0.707·32-s + 0.174·33-s + 0.507·35-s + 1/6·36-s + 0.493·37-s − 0.640·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 830 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 830 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.3667955802\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3667955802\) |
| \(L(\frac{3}{2})\) |
|
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$ | \( ( 1 - T )( 1 + T + p T^{2} ) \) |
| 5 | $C_1$$\times$$C_2$ | \( ( 1 - T )( 1 + 2 T + p T^{2} ) \) |
| 83 | $C_1$$\times$$C_2$ | \( ( 1 + T )( 1 - 4 T + p T^{2} ) \) |
| good | 3 | $D_{4}$ | \( 1 + T + 2 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 7 | $D_{4}$ | \( 1 + 3 T + 6 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 11 | $D_{4}$ | \( 1 + T - 6 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 17 | $C_2^2$ | \( 1 - 3 T + 20 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 + 6 T + 30 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 29 | $D_{4}$ | \( 1 - 3 T + 16 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 - T - 2 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 - 3 T - 8 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 8 T + 30 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 + 6 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 53 | $D_{4}$ | \( 1 - 8 T + 54 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 9 T + 34 T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2$$\times$$C_2$ | \( ( 1 + 2 T + p T^{2} )( 1 + 15 T + p T^{2} ) \) |
| 67 | $D_{4}$ | \( 1 + 6 T - 2 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 6 T - 2 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 73 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
| 79 | $D_{4}$ | \( 1 + 10 T + 126 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 6 T - 14 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 14 T + 154 T^{2} - 14 p T^{3} + p^{2} T^{4} \) |
| 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
−19.5925294770, −19.2572972047, −18.6324562103, −18.2336753703, −17.2975566861, −16.8791468219, −16.4604204725, −15.8035074872, −15.2495014399, −14.4958547433, −13.6363232943, −13.2045450218, −12.6887834577, −11.8952409805, −11.1809919781, −10.4773767688, −9.98752488787, −8.91522349593, −8.35888323784, −7.37090350834, −6.40845350506, −5.67358067082, −4.48414475183, −3.39950407497,
3.39950407497, 4.48414475183, 5.67358067082, 6.40845350506, 7.37090350834, 8.35888323784, 8.91522349593, 9.98752488787, 10.4773767688, 11.1809919781, 11.8952409805, 12.6887834577, 13.2045450218, 13.6363232943, 14.4958547433, 15.2495014399, 15.8035074872, 16.4604204725, 16.8791468219, 17.2975566861, 18.2336753703, 18.6324562103, 19.2572972047, 19.5925294770
