| L(s) = 1 | − 2-s + 4-s − 8·5-s − 8-s + 9-s + 8·10-s − 12·13-s + 16-s − 2·17-s − 18-s − 8·20-s + 38·25-s + 12·26-s − 8·29-s − 32-s + 2·34-s + 36-s − 8·37-s + 8·40-s − 20·41-s − 8·45-s − 10·49-s − 38·50-s − 12·52-s − 4·53-s + 8·58-s − 8·61-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s − 3.57·5-s − 0.353·8-s + 1/3·9-s + 2.52·10-s − 3.32·13-s + 1/4·16-s − 0.485·17-s − 0.235·18-s − 1.78·20-s + 38/5·25-s + 2.35·26-s − 1.48·29-s − 0.176·32-s + 0.342·34-s + 1/6·36-s − 1.31·37-s + 1.26·40-s − 3.12·41-s − 1.19·45-s − 1.42·49-s − 5.37·50-s − 1.66·52-s − 0.549·53-s + 1.05·58-s − 1.02·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 83232 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 83232 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | 2 | $C_1$ | \( 1 + T \) |
| 3 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| 17 | $C_1$ | \( ( 1 + T )^{2} \) |
| good | 5 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 7 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 11 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 73 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 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
−8.951168622566808298087877447389, −8.687931578430379170921838417106, −8.005132056516948045631634947890, −7.56738111269202357438702690211, −7.50224155158573118874050462938, −6.99640780173569532110131873676, −6.63257438834160544258861284685, −5.00716852164944696644803170219, −4.91997035368174263783342884746, −4.36159048011225245066599957337, −3.36779318987556938646346405293, −3.34589616522309311397390467846, −2.09209863939800055422786202913, 0, 0,
2.09209863939800055422786202913, 3.34589616522309311397390467846, 3.36779318987556938646346405293, 4.36159048011225245066599957337, 4.91997035368174263783342884746, 5.00716852164944696644803170219, 6.63257438834160544258861284685, 6.99640780173569532110131873676, 7.50224155158573118874050462938, 7.56738111269202357438702690211, 8.005132056516948045631634947890, 8.687931578430379170921838417106, 8.951168622566808298087877447389
