| L(s) = 1 | − 2-s + 2·3-s + 4-s − 2·6-s − 7-s − 3·8-s − 2·9-s + 4·11-s + 2·12-s + 3·13-s + 14-s + 16-s − 7·17-s + 2·18-s + 7·19-s − 2·21-s − 4·22-s − 23-s − 6·24-s − 6·25-s − 3·26-s − 10·27-s − 28-s − 29-s − 3·31-s + 32-s + 8·33-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1.15·3-s + 1/2·4-s − 0.816·6-s − 0.377·7-s − 1.06·8-s − 2/3·9-s + 1.20·11-s + 0.577·12-s + 0.832·13-s + 0.267·14-s + 1/4·16-s − 1.69·17-s + 0.471·18-s + 1.60·19-s − 0.436·21-s − 0.852·22-s − 0.208·23-s − 1.22·24-s − 6/5·25-s − 0.588·26-s − 1.92·27-s − 0.188·28-s − 0.185·29-s − 0.538·31-s + 0.176·32-s + 1.39·33-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3661 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3661 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.7183587978\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7183587978\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 7 | $C_1$$\times$$C_2$ | \( ( 1 + T )( 1 + p T^{2} ) \) |
| 523 | $C_1$$\times$$C_2$ | \( ( 1 + T )( 1 + 20 T + p T^{2} ) \) |
| good | 2 | $D_{4}$ | \( 1 + T + p T^{3} + p^{2} T^{4} \) |
| 3 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + p T^{2} ) \) |
| 5 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 11 | $C_4$ | \( 1 - 4 T + 6 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 13 | $D_{4}$ | \( 1 - 3 T - 3 p T^{3} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 + 7 T + 32 T^{2} + 7 p T^{3} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 - 7 T + 34 T^{2} - 7 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2$$\times$$C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 29 | $D_{4}$ | \( 1 + T - 8 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 3 T - 18 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 - 2 T - 14 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 7 T + 40 T^{2} + 7 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - 13 T + 114 T^{2} - 13 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 6 T + 70 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2$$\times$$C_2$ | \( ( 1 - 9 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 59 | $D_{4}$ | \( 1 - 2 T + 62 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 6 T - 22 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 4 T + 62 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 3 T + 118 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 73 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 11 T + p T^{2} ) \) |
| 79 | $D_{4}$ | \( 1 + 6 T + 126 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2$$\times$$C_2$ | \( ( 1 - 16 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 89 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 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
−17.9001399225, −17.5142988317, −17.1292585118, −16.2669017847, −15.7991799165, −15.4623486776, −14.7893633794, −14.2842629363, −13.6892568259, −13.5154908046, −12.5375201627, −11.7413178630, −11.4251519946, −10.9845222485, −9.78728074676, −9.33560294007, −8.97374706935, −8.54164397740, −7.82888744063, −6.96807475471, −6.25275854416, −5.63993729563, −4.00972512728, −3.26720242565, −2.27252559043,
2.27252559043, 3.26720242565, 4.00972512728, 5.63993729563, 6.25275854416, 6.96807475471, 7.82888744063, 8.54164397740, 8.97374706935, 9.33560294007, 9.78728074676, 10.9845222485, 11.4251519946, 11.7413178630, 12.5375201627, 13.5154908046, 13.6892568259, 14.2842629363, 14.7893633794, 15.4623486776, 15.7991799165, 16.2669017847, 17.1292585118, 17.5142988317, 17.9001399225
