| L(s) = 1 | + 2-s + 2·3-s − 2·5-s + 2·6-s + 8-s − 2·10-s + 2·11-s + 4·13-s − 4·15-s − 16-s + 6·17-s − 4·19-s + 2·22-s + 2·24-s + 3·25-s + 4·26-s − 2·27-s − 3·29-s − 4·30-s − 8·31-s − 6·32-s + 4·33-s + 6·34-s + 37-s − 4·38-s + 8·39-s − 2·40-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1.15·3-s − 0.894·5-s + 0.816·6-s + 0.353·8-s − 0.632·10-s + 0.603·11-s + 1.10·13-s − 1.03·15-s − 1/4·16-s + 1.45·17-s − 0.917·19-s + 0.426·22-s + 0.408·24-s + 3/5·25-s + 0.784·26-s − 0.384·27-s − 0.557·29-s − 0.730·30-s − 1.43·31-s − 1.06·32-s + 0.696·33-s + 1.02·34-s + 0.164·37-s − 0.648·38-s + 1.28·39-s − 0.316·40-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 17225 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 17225 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.921139994\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.921139994\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 5 | $C_1$ | \( ( 1 + T )^{2} \) |
| 13 | $C_1$$\times$$C_2$ | \( ( 1 + T )( 1 - 5 T + p T^{2} ) \) |
| 53 | $C_1$$\times$$C_2$ | \( ( 1 + T )( 1 - 3 T + p T^{2} ) \) |
| good | 2 | $D_{4}$ | \( 1 - T + T^{2} - p T^{3} + p^{2} T^{4} \) |
| 3 | $D_{4}$ | \( 1 - 2 T + 4 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 7 | $C_2^2$ | \( 1 + 4 T^{2} + p^{2} T^{4} \) |
| 11 | $D_{4}$ | \( 1 - 2 T - 8 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - 6 T + 19 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 + 4 T - T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 + 34 T^{2} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 3 T + 28 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 37 | $D_{4}$ | \( 1 - T + 36 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2^2$ | \( 1 + 37 T^{2} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 6 T + T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 6 T + 28 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 8 T + 16 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 2 T - 70 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 6 T + 13 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2^2$ | \( 1 - 86 T^{2} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 14 T + 134 T^{2} - 14 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 2 T + 96 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 8 T + 103 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 9 T + 70 T^{2} + 9 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 7 T + 114 T^{2} - 7 p T^{3} + p^{2} T^{4} \) |
| 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
−15.9005681813, −15.1220390524, −14.8220972911, −14.5522992963, −14.0418695399, −13.6168353485, −13.1912729027, −12.5776849955, −12.2532117598, −11.4936853759, −11.0386318046, −10.7263734601, −9.74455684712, −9.29772690230, −8.65700324845, −8.25630547803, −7.84421936744, −7.10727560163, −6.49854577793, −5.65654974526, −4.96017139677, −4.03994384745, −3.66794976401, −3.09182838678, −1.77381834964,
1.77381834964, 3.09182838678, 3.66794976401, 4.03994384745, 4.96017139677, 5.65654974526, 6.49854577793, 7.10727560163, 7.84421936744, 8.25630547803, 8.65700324845, 9.29772690230, 9.74455684712, 10.7263734601, 11.0386318046, 11.4936853759, 12.2532117598, 12.5776849955, 13.1912729027, 13.6168353485, 14.0418695399, 14.5522992963, 14.8220972911, 15.1220390524, 15.9005681813
