| L(s) = 1 | + 2·3-s − 2·7-s + 2·9-s − 4·21-s + 2·23-s + 2·27-s − 2·43-s + 2·47-s + 2·49-s − 4·63-s − 2·67-s + 4·69-s + 3·81-s − 2·83-s + 4·101-s − 2·103-s + 2·107-s − 2·121-s + 127-s − 4·129-s + 131-s + 137-s + 139-s + 4·141-s + 4·147-s + 149-s + 151-s + ⋯ |
| L(s) = 1 | + 2·3-s − 2·7-s + 2·9-s − 4·21-s + 2·23-s + 2·27-s − 2·43-s + 2·47-s + 2·49-s − 4·63-s − 2·67-s + 4·69-s + 3·81-s − 2·83-s + 4·101-s − 2·103-s + 2·107-s − 2·121-s + 127-s − 4·129-s + 131-s + 137-s + 139-s + 4·141-s + 4·147-s + 149-s + 151-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 640000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 640000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(1.447579374\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.447579374\) |
| \(L(1)\) |
|
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 | | \( 1 \) |
| good | 3 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 + T^{2} ) \) |
| 7 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{2}( 1 + T^{2} ) \) |
| 11 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 + T^{4} \) |
| 17 | $C_2^2$ | \( 1 + T^{4} \) |
| 19 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 23 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 + T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 + T^{4} \) |
| 41 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 43 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{2}( 1 + T^{2} ) \) |
| 47 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 + T^{2} ) \) |
| 53 | $C_2^2$ | \( 1 + T^{4} \) |
| 59 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 61 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 67 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{2}( 1 + T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 + T^{4} \) |
| 79 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 83 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{2}( 1 + T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 + T^{4} \) |
| 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
−10.27436645597235270369076056873, −10.25418350284940333018242297933, −9.712318834154996844364446445339, −9.235042093885249695731507453882, −9.035576405082033067896923317804, −8.602158146563770624393973871475, −8.472426697580318999022771052851, −7.57101883992844295644297705453, −7.33705878273929921372252718273, −6.99530368165946003159275129257, −6.32662121816747505270602289693, −6.18178692110613313626959723916, −5.27022331743516471751462930084, −4.78912913688693580623298635546, −4.06560687717947572477017256657, −3.48879543380692432366847665689, −3.21439990882608355955517968670, −2.77337933298861100698574492004, −2.37520475548774116647543465138, −1.27596552012151008585594330858,
1.27596552012151008585594330858, 2.37520475548774116647543465138, 2.77337933298861100698574492004, 3.21439990882608355955517968670, 3.48879543380692432366847665689, 4.06560687717947572477017256657, 4.78912913688693580623298635546, 5.27022331743516471751462930084, 6.18178692110613313626959723916, 6.32662121816747505270602289693, 6.99530368165946003159275129257, 7.33705878273929921372252718273, 7.57101883992844295644297705453, 8.472426697580318999022771052851, 8.602158146563770624393973871475, 9.035576405082033067896923317804, 9.235042093885249695731507453882, 9.712318834154996844364446445339, 10.25418350284940333018242297933, 10.27436645597235270369076056873
