L(s) = 1 | − 5-s + 2·9-s − 4·13-s − 4·17-s − 2·25-s − 4·29-s + 4·37-s + 4·41-s − 2·45-s + 2·49-s − 4·53-s + 20·61-s + 4·65-s + 12·73-s − 5·81-s + 4·85-s + 4·89-s − 4·97-s − 4·101-s + 4·109-s − 28·113-s − 8·117-s + 10·121-s + 10·125-s + 127-s + 131-s + 137-s + ⋯ |
L(s) = 1 | − 0.447·5-s + 2/3·9-s − 1.10·13-s − 0.970·17-s − 2/5·25-s − 0.742·29-s + 0.657·37-s + 0.624·41-s − 0.298·45-s + 2/7·49-s − 0.549·53-s + 2.56·61-s + 0.496·65-s + 1.40·73-s − 5/9·81-s + 0.433·85-s + 0.423·89-s − 0.406·97-s − 0.398·101-s + 0.383·109-s − 2.63·113-s − 0.739·117-s + 0.909·121-s + 0.894·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2560 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2560 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.6563117485\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.6563117485\) |
\(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 5 | $C_1$$\times$$C_2$ | \( ( 1 - T )( 1 + 2 T + p T^{2} ) \) |
good | 3 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 7 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 11 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 17 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 19 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 - 34 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 37 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 41 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 43 | $C_2^2$ | \( 1 + 62 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 59 | $C_2^2$ | \( 1 + 38 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$$\times$$C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 - 6 T + p T^{2} ) \) |
| 67 | $C_2^2$ | \( 1 - 18 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2^2$ | \( 1 + 46 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) |
| 79 | $C_2^2$ | \( 1 - 34 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2^2$ | \( 1 + 126 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−12.95976264669693540025589192078, −12.48367628325372425340654340374, −11.80372322771740053448933326215, −11.23611624623193562674582428404, −10.65505975474798628359860403767, −9.774070737967069714239308188778, −9.480468180241242559684660262579, −8.555161022741242013168039523731, −7.81855471798383057814968612224, −7.20447908396518733838295689153, −6.57158847385268159714087019173, −5.49604786544735417724275437402, −4.59656659721016932035456370597, −3.83789795478194203908952192251, −2.35673748189235633398390138135,
2.35673748189235633398390138135, 3.83789795478194203908952192251, 4.59656659721016932035456370597, 5.49604786544735417724275437402, 6.57158847385268159714087019173, 7.20447908396518733838295689153, 7.81855471798383057814968612224, 8.555161022741242013168039523731, 9.480468180241242559684660262579, 9.774070737967069714239308188778, 10.65505975474798628359860403767, 11.23611624623193562674582428404, 11.80372322771740053448933326215, 12.48367628325372425340654340374, 12.95976264669693540025589192078