| L(s) = 1 | − 2-s − 3-s − 4·5-s + 6-s − 2·7-s + 8-s − 2·9-s + 4·10-s − 4·11-s + 2·13-s + 2·14-s + 4·15-s − 16-s + 2·17-s + 2·18-s + 2·21-s + 4·22-s − 2·23-s − 24-s + 3·25-s − 2·26-s + 5·27-s + 6·29-s − 4·30-s − 4·31-s + 4·33-s − 2·34-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s − 1.78·5-s + 0.408·6-s − 0.755·7-s + 0.353·8-s − 2/3·9-s + 1.26·10-s − 1.20·11-s + 0.554·13-s + 0.534·14-s + 1.03·15-s − 1/4·16-s + 0.485·17-s + 0.471·18-s + 0.436·21-s + 0.852·22-s − 0.417·23-s − 0.204·24-s + 3/5·25-s − 0.392·26-s + 0.962·27-s + 1.11·29-s − 0.730·30-s − 0.718·31-s + 0.696·33-s − 0.342·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3564 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3564 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | $C_2$ | \( 1 + T + T^{2} \) |
| 3 | $C_2$ | \( 1 + T + p T^{2} \) |
| 11 | $C_1$$\times$$C_2$ | \( ( 1 + T )( 1 + 3 T + p T^{2} ) \) |
| good | 5 | $C_2$$\times$$C_2$ | \( ( 1 + T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 7 | $D_{4}$ | \( 1 + 2 T + 3 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 13 | $D_{4}$ | \( 1 - 2 T - 7 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 - 2 T - 5 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 + 25 T^{2} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 + 2 T + 7 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2$$\times$$C_2$ | \( ( 1 - 9 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 31 | $D_{4}$ | \( 1 + 4 T + 29 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 4 T + 38 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2$$\times$$C_2$ | \( ( 1 - 9 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 43 | $D_{4}$ | \( 1 + 3 T + 10 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2$$\times$$C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) |
| 53 | $D_{4}$ | \( 1 + 4 T + 82 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 3 T + 82 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 25 T^{2} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 3 T + 58 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2^2$ | \( 1 + 118 T^{2} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 4 T - 31 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 4 T - 9 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 4 T - p T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 8 T + 142 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 3 T - 56 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 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
−18.3156652628, −17.7379356222, −17.3209720789, −16.4444005174, −16.2411763542, −15.8542232003, −15.4087015597, −14.7548406130, −13.9830458970, −13.4220604745, −12.6595257910, −12.2091702465, −11.6824337466, −11.1059082761, −10.6337197948, −10.0165033949, −9.26396487621, −8.29422178529, −8.17525606727, −7.48063533198, −6.66376602863, −5.79237034347, −4.93739784561, −3.89603464685, −3.04992171089, 0,
3.04992171089, 3.89603464685, 4.93739784561, 5.79237034347, 6.66376602863, 7.48063533198, 8.17525606727, 8.29422178529, 9.26396487621, 10.0165033949, 10.6337197948, 11.1059082761, 11.6824337466, 12.2091702465, 12.6595257910, 13.4220604745, 13.9830458970, 14.7548406130, 15.4087015597, 15.8542232003, 16.2411763542, 16.4444005174, 17.3209720789, 17.7379356222, 18.3156652628
