| L(s) = 1 | + 2-s + 3-s − 4-s + 6-s − 7-s − 8-s + 9-s − 2·11-s − 12-s − 3·13-s − 14-s + 3·16-s + 17-s + 18-s − 5·19-s − 21-s − 2·22-s − 23-s − 24-s − 4·25-s − 3·26-s + 4·27-s + 28-s − 29-s + 7·31-s + 3·32-s − 2·33-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.577·3-s − 1/2·4-s + 0.408·6-s − 0.377·7-s − 0.353·8-s + 1/3·9-s − 0.603·11-s − 0.288·12-s − 0.832·13-s − 0.267·14-s + 3/4·16-s + 0.242·17-s + 0.235·18-s − 1.14·19-s − 0.218·21-s − 0.426·22-s − 0.208·23-s − 0.204·24-s − 4/5·25-s − 0.588·26-s + 0.769·27-s + 0.188·28-s − 0.185·29-s + 1.25·31-s + 0.530·32-s − 0.348·33-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4418 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4418 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.086460387\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.086460387\) |
| \(L(\frac{3}{2})\) |
|
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$$\times$$C_2$ | \( ( 1 - T )( 1 + p T^{2} ) \) |
| 47 | $C_2$ | \( 1 + p T^{2} \) |
| good | 3 | $C_2$$\times$$C_2$ | \( ( 1 - p T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 5 | $C_2^2$ | \( 1 + 4 T^{2} + p^{2} T^{4} \) |
| 7 | $C_4$ | \( 1 + T + 5 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 11 | $D_{4}$ | \( 1 + 2 T + 4 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2$$\times$$C_2$ | \( ( 1 - T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 17 | $C_2$$\times$$C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 19 | $C_2$$\times$$C_2$ | \( ( 1 + T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 23 | $D_{4}$ | \( 1 + T + 25 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + T + 22 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 37 | $D_{4}$ | \( 1 + 2 T + 48 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 - 3 T + 61 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - 5 T + 8 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 4 T + 58 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 6 T + 94 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - T + 48 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 12 T + 160 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 5 T + 37 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 7 T + 39 T^{2} - 7 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 83 | $D_{4}$ | \( 1 - 2 T + 34 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + T + 73 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 3 T + 115 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 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
−17.6584161869, −17.1013895911, −16.7761260868, −15.8262056228, −15.6249361979, −14.9586755789, −14.4081732886, −14.0514266016, −13.5580019143, −12.9220497275, −12.5770885086, −12.1714010082, −11.2475477337, −10.5035074410, −9.94758037096, −9.53776853723, −8.69047778993, −8.08684358755, −7.57205103600, −6.62139632297, −5.87151509282, −4.95875964272, −4.37456514466, −3.48289742797, −2.43594889399,
2.43594889399, 3.48289742797, 4.37456514466, 4.95875964272, 5.87151509282, 6.62139632297, 7.57205103600, 8.08684358755, 8.69047778993, 9.53776853723, 9.94758037096, 10.5035074410, 11.2475477337, 12.1714010082, 12.5770885086, 12.9220497275, 13.5580019143, 14.0514266016, 14.4081732886, 14.9586755789, 15.6249361979, 15.8262056228, 16.7761260868, 17.1013895911, 17.6584161869
