| L(s) = 1 | − 2·2-s − 3·3-s + 4-s − 3·5-s + 6·6-s + 3·9-s + 6·10-s − 3·11-s − 3·12-s + 5·13-s + 9·15-s + 16-s − 6·18-s − 8·19-s − 3·20-s + 6·22-s + 23-s + 4·25-s − 10·26-s − 4·29-s − 18·30-s − 31-s + 2·32-s + 9·33-s + 3·36-s − 7·37-s + 16·38-s + ⋯ |
| L(s) = 1 | − 1.41·2-s − 1.73·3-s + 1/2·4-s − 1.34·5-s + 2.44·6-s + 9-s + 1.89·10-s − 0.904·11-s − 0.866·12-s + 1.38·13-s + 2.32·15-s + 1/4·16-s − 1.41·18-s − 1.83·19-s − 0.670·20-s + 1.27·22-s + 0.208·23-s + 4/5·25-s − 1.96·26-s − 0.742·29-s − 3.28·30-s − 0.179·31-s + 0.353·32-s + 1.56·33-s + 1/2·36-s − 1.15·37-s + 2.59·38-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1091 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1091 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 1091 | $C_1$$\times$$C_2$ | \( ( 1 - T )( 1 + 21 T + p T^{2} ) \) |
| good | 2 | $D_{4}$ | \( 1 + p T + 3 T^{2} + p^{2} T^{3} + p^{2} T^{4} \) |
| 3 | $C_2$ | \( ( 1 + p T^{2} )( 1 + p T + p T^{2} ) \) |
| 5 | $D_{4}$ | \( 1 + 3 T + p T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 7 | $C_2^2$ | \( 1 + 6 T^{2} + p^{2} T^{4} \) |
| 11 | $D_{4}$ | \( 1 + 3 T + 10 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 13 | $D_{4}$ | \( 1 - 5 T + 19 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 + 8 T + 52 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2$$\times$$C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 29 | $D_{4}$ | \( 1 + 4 T + 18 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + T - 42 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 7 T + 58 T^{2} + 7 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 47 | $D_{4}$ | \( 1 + 2 T + 60 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 12 T + 134 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 3 T + 83 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 67 | $D_{4}$ | \( 1 - 7 T + 109 T^{2} - 7 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2^2$ | \( 1 + 54 T^{2} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - T + 85 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 3 T - 45 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - T - 86 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 2 T - 24 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 2 T - 2 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 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
−19.8087339679, −19.2927698249, −18.7588261401, −18.4252657333, −17.9325643653, −17.3940434538, −16.8904747008, −16.4983732150, −15.8702950967, −15.3577164109, −14.8289717289, −13.5671344301, −12.9058210635, −12.2374732596, −11.5873611136, −11.0546519721, −10.7201337516, −10.0391382416, −8.86324040146, −8.45331855624, −7.82014337708, −6.75157269231, −5.99604896663, −5.03981995913, −3.82540183589, 0,
3.82540183589, 5.03981995913, 5.99604896663, 6.75157269231, 7.82014337708, 8.45331855624, 8.86324040146, 10.0391382416, 10.7201337516, 11.0546519721, 11.5873611136, 12.2374732596, 12.9058210635, 13.5671344301, 14.8289717289, 15.3577164109, 15.8702950967, 16.4983732150, 16.8904747008, 17.3940434538, 17.9325643653, 18.4252657333, 18.7588261401, 19.2927698249, 19.8087339679
