| L(s) = 1 | + 2-s + 2·3-s − 4-s + 2·6-s − 3·8-s + 9-s + 2·11-s − 2·12-s − 6·13-s − 16-s + 18-s + 2·22-s + 8·23-s − 6·24-s − 6·25-s − 6·26-s − 4·27-s + 5·32-s + 4·33-s − 36-s − 4·37-s − 12·39-s − 2·44-s + 8·46-s + 11·47-s − 2·48-s − 2·49-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1.15·3-s − 1/2·4-s + 0.816·6-s − 1.06·8-s + 1/3·9-s + 0.603·11-s − 0.577·12-s − 1.66·13-s − 1/4·16-s + 0.235·18-s + 0.426·22-s + 1.66·23-s − 1.22·24-s − 6/5·25-s − 1.17·26-s − 0.769·27-s + 0.883·32-s + 0.696·33-s − 1/6·36-s − 0.657·37-s − 1.92·39-s − 0.301·44-s + 1.17·46-s + 1.60·47-s − 0.288·48-s − 2/7·49-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6768 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6768 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.400209051\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.400209051\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 - T + p T^{2} \) |
| 3 | $C_2$ | \( 1 - 2 T + p T^{2} \) |
| 47 | $C_1$$\times$$C_2$ | \( ( 1 + T )( 1 - 12 T + p T^{2} ) \) |
| good | 5 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 7 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + p T^{2} ) \) |
| 13 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 17 | $C_2^2$ | \( 1 + 10 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 - 34 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + p T^{2} ) \) |
| 29 | $C_2^2$ | \( 1 - 50 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2^2$ | \( 1 + 10 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 41 | $C_2^2$ | \( 1 + 30 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 - 42 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 58 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$$\times$$C_2$ | \( ( 1 + 6 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 61 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 - 4 T + p T^{2} ) \) |
| 67 | $C_2^2$ | \( 1 + 46 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + p T^{2} ) \) |
| 73 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 79 | $C_2^2$ | \( 1 + 38 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
| 89 | $C_2^2$ | \( 1 + 130 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 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
−12.06348536069737901869235470893, −11.61136302917026346955939133989, −10.81114434133019088285592474072, −9.900453461440815959072598993825, −9.538216544202169346491531105047, −9.004335937943108396768132310623, −8.574080056020246049951316049876, −7.70526516823376453528333355376, −7.25263676501326636619443467606, −6.35722883469283730551194670504, −5.44125526310033539921062120439, −4.82450904845450854211338425341, −3.98309004135621702286649354693, −3.22490292678189459921069600882, −2.36053813804726347490900901412,
2.36053813804726347490900901412, 3.22490292678189459921069600882, 3.98309004135621702286649354693, 4.82450904845450854211338425341, 5.44125526310033539921062120439, 6.35722883469283730551194670504, 7.25263676501326636619443467606, 7.70526516823376453528333355376, 8.574080056020246049951316049876, 9.004335937943108396768132310623, 9.538216544202169346491531105047, 9.900453461440815959072598993825, 10.81114434133019088285592474072, 11.61136302917026346955939133989, 12.06348536069737901869235470893
