| L(s) = 1 | − 2-s + 4-s − 3·5-s + 2·7-s − 8-s + 9-s + 3·10-s − 2·14-s + 16-s − 18-s − 2·19-s − 3·20-s − 25-s + 2·28-s − 32-s − 6·35-s + 36-s − 8·37-s + 2·38-s + 3·40-s + 10·43-s − 3·45-s − 3·49-s + 50-s + 18·53-s − 2·56-s + 2·63-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s − 1.34·5-s + 0.755·7-s − 0.353·8-s + 1/3·9-s + 0.948·10-s − 0.534·14-s + 1/4·16-s − 0.235·18-s − 0.458·19-s − 0.670·20-s − 1/5·25-s + 0.377·28-s − 0.176·32-s − 1.01·35-s + 1/6·36-s − 1.31·37-s + 0.324·38-s + 0.474·40-s + 1.52·43-s − 0.447·45-s − 3/7·49-s + 0.141·50-s + 2.47·53-s − 0.267·56-s + 0.251·63-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 189728 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 189728 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.8567179621\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8567179621\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( 1 + T \) |
| 7 | $C_2$ | \( 1 - 2 T + p T^{2} \) |
| 11 | $C_2$ | \( 1 + p T^{2} \) |
| good | 3 | $C_2^2$ | \( 1 - T^{2} + p^{2} T^{4} \) |
| 5 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 13 | $C_2^2$ | \( 1 - 20 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - 5 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$$\times$$C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 23 | $C_2^2$ | \( 1 - 38 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 + 28 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2^2$ | \( 1 - 14 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 41 | $C_2^2$ | \( 1 + 19 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) |
| 47 | $C_2^2$ | \( 1 - 8 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 - 6 T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 61 | $C_2^2$ | \( 1 - 86 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 - 119 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2^2$ | \( 1 + 100 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2^2$ | \( 1 - 11 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$$\times$$C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 - 8 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 - 9 T + p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 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
−9.010091074976382727914106250529, −8.631734377972301495125464310511, −8.048779860403113945093150558366, −7.80943641274813525276383170126, −7.41818618677176957057908440103, −6.88418165117100567790832155048, −6.39800033659735776883932213571, −5.65431080299592880787597045483, −5.12621462035049185692507860629, −4.45772074828101566475956002221, −3.89684374746954086118612834103, −3.52346778658138323025914241320, −2.48352131966472328989125101835, −1.81342033887732970486304429798, −0.67258321115543357013059760564,
0.67258321115543357013059760564, 1.81342033887732970486304429798, 2.48352131966472328989125101835, 3.52346778658138323025914241320, 3.89684374746954086118612834103, 4.45772074828101566475956002221, 5.12621462035049185692507860629, 5.65431080299592880787597045483, 6.39800033659735776883932213571, 6.88418165117100567790832155048, 7.41818618677176957057908440103, 7.80943641274813525276383170126, 8.048779860403113945093150558366, 8.631734377972301495125464310511, 9.010091074976382727914106250529
