| L(s) = 1 | + 2-s − 3·5-s − 8-s − 3·9-s − 3·10-s + 6·11-s + 2·13-s − 16-s − 12·17-s − 3·18-s + 14·19-s + 6·22-s − 3·23-s + 5·25-s + 2·26-s − 6·29-s + 2·31-s − 12·34-s + 4·37-s + 14·38-s + 3·40-s − 2·43-s + 9·45-s − 3·46-s + 5·50-s + 12·53-s − 18·55-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 1.34·5-s − 0.353·8-s − 9-s − 0.948·10-s + 1.80·11-s + 0.554·13-s − 1/4·16-s − 2.91·17-s − 0.707·18-s + 3.21·19-s + 1.27·22-s − 0.625·23-s + 25-s + 0.392·26-s − 1.11·29-s + 0.359·31-s − 2.05·34-s + 0.657·37-s + 2.27·38-s + 0.474·40-s − 0.304·43-s + 1.34·45-s − 0.442·46-s + 0.707·50-s + 1.64·53-s − 2.42·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 777924 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 777924 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.683159637\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.683159637\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 - T + T^{2} \) |
| 3 | $C_2$ | \( 1 + p T^{2} \) |
| 7 | | \( 1 \) |
| good | 5 | $C_2^2$ | \( 1 + 3 T + 4 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 6 T + 25 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - 7 T + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 + 3 T - 14 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 + 6 T + 7 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2^2$ | \( 1 - 2 T - 27 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 + 2 T - 39 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 5 T - 36 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 + 8 T - 3 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 79 | $C_2^2$ | \( 1 + 5 T - 54 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2^2$ | \( 1 - 12 T + 61 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 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
−10.59694363920667051924635489392, −9.736000199304687786589764240958, −9.290633417278735376827883738495, −9.117445548242253407093474649899, −8.789726955683827053751237734212, −8.175858267806827663603888974826, −7.899321443447152261510493070964, −7.19623098128942279082882958680, −6.96945011160415676378517493244, −6.42009486056554632692770233050, −6.08502832549559124301236003416, −5.33142328398286281969769102218, −5.14872375829298971321837624691, −4.24839739764351505823086831606, −4.06336083359052502233310536445, −3.71897462971167064660627416686, −3.09115944200524700898693665764, −2.55538912827991486450575441755, −1.54597252019072772210147034697, −0.58195978553675455293024134787,
0.58195978553675455293024134787, 1.54597252019072772210147034697, 2.55538912827991486450575441755, 3.09115944200524700898693665764, 3.71897462971167064660627416686, 4.06336083359052502233310536445, 4.24839739764351505823086831606, 5.14872375829298971321837624691, 5.33142328398286281969769102218, 6.08502832549559124301236003416, 6.42009486056554632692770233050, 6.96945011160415676378517493244, 7.19623098128942279082882958680, 7.899321443447152261510493070964, 8.175858267806827663603888974826, 8.789726955683827053751237734212, 9.117445548242253407093474649899, 9.290633417278735376827883738495, 9.736000199304687786589764240958, 10.59694363920667051924635489392
