| L(s) = 1 | − 3-s − 2·5-s − 4·7-s + 9-s + 4·11-s + 2·15-s + 2·17-s + 4·19-s + 4·21-s − 8·23-s − 6·25-s − 27-s + 6·29-s + 4·31-s − 4·33-s + 8·35-s − 4·37-s − 6·41-s + 12·43-s − 2·45-s + 2·49-s − 2·51-s − 2·53-s − 8·55-s − 4·57-s + 4·59-s + 12·61-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.894·5-s − 1.51·7-s + 1/3·9-s + 1.20·11-s + 0.516·15-s + 0.485·17-s + 0.917·19-s + 0.872·21-s − 1.66·23-s − 6/5·25-s − 0.192·27-s + 1.11·29-s + 0.718·31-s − 0.696·33-s + 1.35·35-s − 0.657·37-s − 0.937·41-s + 1.82·43-s − 0.298·45-s + 2/7·49-s − 0.280·51-s − 0.274·53-s − 1.07·55-s − 0.529·57-s + 0.520·59-s + 1.53·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 864 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 864 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.3779784563\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3779784563\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 3 | $C_1$ | \( 1 + T \) |
| good | 5 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 7 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 11 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + p T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 17 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + p T^{2} ) \) |
| 19 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 23 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 29 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + p T^{2} ) \) |
| 31 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 37 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 41 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 43 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 - 4 T + p T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 53 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 59 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + p T^{2} ) \) |
| 61 | $C_2$$\times$$C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 + 4 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
| 71 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + p T^{2} ) \) |
| 73 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 79 | $C_2$$\times$$C_2$ | \( ( 1 + 4 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 83 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 89 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) |
| 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
−19.8934316645, −19.7485854378, −19.1551537949, −18.8585099876, −17.6873258241, −17.6707519500, −16.6272610009, −16.2503860035, −15.6969681316, −15.4732516475, −14.1819882616, −13.9963410512, −13.0105598262, −12.3172568607, −11.7743766738, −11.5820474614, −10.1744110310, −9.96467191573, −9.11342494550, −8.09899069409, −7.26646731082, −6.42897107073, −5.80268955255, −4.25303028693, −3.44334336791,
3.44334336791, 4.25303028693, 5.80268955255, 6.42897107073, 7.26646731082, 8.09899069409, 9.11342494550, 9.96467191573, 10.1744110310, 11.5820474614, 11.7743766738, 12.3172568607, 13.0105598262, 13.9963410512, 14.1819882616, 15.4732516475, 15.6969681316, 16.2503860035, 16.6272610009, 17.6707519500, 17.6873258241, 18.8585099876, 19.1551537949, 19.7485854378, 19.8934316645
