| L(s) = 1 | + 2-s − 3·3-s + 4-s − 3·6-s − 7-s + 8-s + 3·9-s − 2·11-s − 3·12-s − 3·13-s − 14-s + 16-s + 3·18-s + 3·19-s + 3·21-s − 2·22-s + 11·23-s − 3·24-s − 2·25-s − 3·26-s − 28-s + 4·29-s − 3·31-s + 32-s + 6·33-s + 3·36-s − 8·37-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 1.73·3-s + 1/2·4-s − 1.22·6-s − 0.377·7-s + 0.353·8-s + 9-s − 0.603·11-s − 0.866·12-s − 0.832·13-s − 0.267·14-s + 1/4·16-s + 0.707·18-s + 0.688·19-s + 0.654·21-s − 0.426·22-s + 2.29·23-s − 0.612·24-s − 2/5·25-s − 0.588·26-s − 0.188·28-s + 0.742·29-s − 0.538·31-s + 0.176·32-s + 1.04·33-s + 1/2·36-s − 1.31·37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1136 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1136 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.4813109863\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4813109863\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( 1 - T \) |
| 71 | $C_1$$\times$$C_2$ | \( ( 1 + T )( 1 - 16 T + p T^{2} ) \) |
| good | 3 | $C_2$ | \( ( 1 + p T^{2} )( 1 + p T + p T^{2} ) \) |
| 5 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 7 | $D_{4}$ | \( 1 + T - 2 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 11 | $D_{4}$ | \( 1 + 2 T - 6 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 13 | $D_{4}$ | \( 1 + 3 T + 12 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 - 3 T + 6 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 - 3 T + p T^{2} ) \) |
| 29 | $D_{4}$ | \( 1 - 4 T + 34 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 3 T + 6 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 8 T + 34 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 - 4 T - 2 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + T - 18 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2$$\times$$C_2$ | \( ( 1 - T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 59 | $D_{4}$ | \( 1 + 2 T - 74 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 67 | $D_{4}$ | \( 1 + 2 T - 6 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 7 T + 88 T^{2} + 7 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2$$\times$$C_2$ | \( ( 1 + 4 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 83 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
| 89 | $D_{4}$ | \( 1 - 3 T + 76 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 8 T + 166 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 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
−19.7780195586, −19.2226014675, −18.6506630456, −17.9257627649, −17.3843547070, −16.9660115899, −16.5873603209, −15.7518239404, −15.5350039380, −14.6291906164, −14.0544554773, −13.1529084742, −12.7377644460, −12.1533915310, −11.4771625556, −11.1238197043, −10.4161290830, −9.74174793918, −8.72496589330, −7.44459689065, −6.89816653273, −5.99376615586, −5.27528562039, −4.79125828888, −3.10049324065,
3.10049324065, 4.79125828888, 5.27528562039, 5.99376615586, 6.89816653273, 7.44459689065, 8.72496589330, 9.74174793918, 10.4161290830, 11.1238197043, 11.4771625556, 12.1533915310, 12.7377644460, 13.1529084742, 14.0544554773, 14.6291906164, 15.5350039380, 15.7518239404, 16.5873603209, 16.9660115899, 17.3843547070, 17.9257627649, 18.6506630456, 19.2226014675, 19.7780195586
