| L(s) = 1 | − 2-s − 2·3-s + 4-s − 4·5-s + 2·6-s − 2·7-s − 3·8-s + 4·9-s + 4·10-s − 11-s − 2·12-s − 2·13-s + 2·14-s + 8·15-s + 16-s − 2·17-s − 4·18-s − 3·19-s − 4·20-s + 4·21-s + 22-s + 3·23-s + 6·24-s + 8·25-s + 2·26-s − 5·27-s − 2·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1.15·3-s + 1/2·4-s − 1.78·5-s + 0.816·6-s − 0.755·7-s − 1.06·8-s + 4/3·9-s + 1.26·10-s − 0.301·11-s − 0.577·12-s − 0.554·13-s + 0.534·14-s + 2.06·15-s + 1/4·16-s − 0.485·17-s − 0.942·18-s − 0.688·19-s − 0.894·20-s + 0.872·21-s + 0.213·22-s + 0.625·23-s + 1.22·24-s + 8/5·25-s + 0.392·26-s − 0.962·27-s − 0.377·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3135 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3135 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | $C_1$$\times$$C_2$ | \( ( 1 - T )( 1 + p T + p T^{2} ) \) |
| 5 | $C_1$$\times$$C_2$ | \( ( 1 + T )( 1 + 3 T + p T^{2} ) \) |
| 11 | $C_1$$\times$$C_2$ | \( ( 1 - T )( 1 + 2 T + p T^{2} ) \) |
| 19 | $C_1$$\times$$C_2$ | \( ( 1 - T )( 1 + 4 T + p T^{2} ) \) |
| good | 2 | $D_{4}$ | \( 1 + T + p T^{3} + p^{2} T^{4} \) |
| 7 | $D_{4}$ | \( 1 + 2 T + 10 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 13 | $D_{4}$ | \( 1 + 2 T + 6 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 17 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 23 | $D_{4}$ | \( 1 - 3 T - 2 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 - 2 T - 2 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 - T + 30 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 - 5 T + 26 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 4 T + 34 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - 4 T + 50 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2$$\times$$C_2$ | \( ( 1 + 4 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 53 | $C_2^2$ | \( 1 + 54 T^{2} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + T + 10 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 12 T + 78 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + T - 50 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 3 T - 50 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 10 T + 114 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 12 T + 110 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 4 T + 110 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 11 T + 70 T^{2} + 11 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 3 T + 34 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 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
−18.5281182613, −17.8869437856, −17.4372728068, −16.8620817168, −16.1337828184, −16.0351721326, −15.5606967613, −15.0065455320, −14.5645497814, −13.1578674561, −12.8873901792, −12.2634201667, −11.7398571627, −11.3246350041, −10.8906200207, −10.0127896862, −9.60450434480, −8.61436974896, −8.12286474353, −7.19748536868, −6.79312508349, −6.13684538177, −4.92309308099, −4.17631003464, −3.00229260477, 0,
3.00229260477, 4.17631003464, 4.92309308099, 6.13684538177, 6.79312508349, 7.19748536868, 8.12286474353, 8.61436974896, 9.60450434480, 10.0127896862, 10.8906200207, 11.3246350041, 11.7398571627, 12.2634201667, 12.8873901792, 13.1578674561, 14.5645497814, 15.0065455320, 15.5606967613, 16.0351721326, 16.1337828184, 16.8620817168, 17.4372728068, 17.8869437856, 18.5281182613
