| L(s) = 1 | − 2·5-s − 4·11-s + 2·13-s − 4·17-s + 8·19-s + 8·23-s + 5·25-s + 6·29-s + 8·31-s + 12·37-s − 6·41-s + 4·43-s + 7·49-s + 4·53-s + 8·55-s − 4·59-s + 2·61-s − 4·65-s − 4·67-s + 16·71-s + 20·73-s − 8·79-s + 4·83-s + 8·85-s + 12·89-s − 16·95-s − 2·97-s + ⋯ |
| L(s) = 1 | − 0.894·5-s − 1.20·11-s + 0.554·13-s − 0.970·17-s + 1.83·19-s + 1.66·23-s + 25-s + 1.11·29-s + 1.43·31-s + 1.97·37-s − 0.937·41-s + 0.609·43-s + 49-s + 0.549·53-s + 1.07·55-s − 0.520·59-s + 0.256·61-s − 0.496·65-s − 0.488·67-s + 1.89·71-s + 2.34·73-s − 0.900·79-s + 0.439·83-s + 0.867·85-s + 1.27·89-s − 1.64·95-s − 0.203·97-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1679616 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1679616 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.066915439\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.066915439\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 3 | | \( 1 \) |
| good | 5 | $C_2^2$ | \( 1 + 2 T - T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 7 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 + 4 T + 5 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 8 T + 41 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 6 T + 7 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2^2$ | \( 1 - 8 T + 33 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 + 6 T - 5 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 - 4 T - 27 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 + 4 T - 43 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 2 T - 57 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 + 4 T - 51 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 79 | $C_2^2$ | \( 1 + 8 T - 15 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2^2$ | \( 1 - 4 T - 67 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 + 2 T - 93 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 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
−9.678155260090871653197900647503, −9.568509157612770623365410196542, −8.962713037240924111840986637938, −8.624330938881923500627405054767, −8.052258597447369362397200679185, −8.029743302419695908881674794434, −7.37570575810996941488469323217, −7.14068162813448002061368247246, −6.48473118501311957369981862663, −6.37222554958022987249871146321, −5.46895078929398705798988423245, −5.24263134539222568747718517582, −4.70111081119873433260608956072, −4.44243641987049034307585558448, −3.71220130776936427451651741337, −3.22829099708825179783608637782, −2.69782398504300168800135679028, −2.42408525011077101508870871840, −1.07256321966832021287789734510, −0.76061982034735608741945139283,
0.76061982034735608741945139283, 1.07256321966832021287789734510, 2.42408525011077101508870871840, 2.69782398504300168800135679028, 3.22829099708825179783608637782, 3.71220130776936427451651741337, 4.44243641987049034307585558448, 4.70111081119873433260608956072, 5.24263134539222568747718517582, 5.46895078929398705798988423245, 6.37222554958022987249871146321, 6.48473118501311957369981862663, 7.14068162813448002061368247246, 7.37570575810996941488469323217, 8.029743302419695908881674794434, 8.052258597447369362397200679185, 8.624330938881923500627405054767, 8.962713037240924111840986637938, 9.568509157612770623365410196542, 9.678155260090871653197900647503
