| L(s) = 1 | − 2·2-s − 2·3-s + 3·4-s + 4·6-s + 2·7-s − 4·8-s + 9-s − 6·12-s − 4·14-s + 5·16-s − 2·18-s + 2·19-s − 4·21-s + 8·24-s − 10·25-s + 4·27-s + 6·28-s − 12·29-s − 6·32-s + 3·36-s − 4·38-s + 12·41-s + 8·42-s + 16·43-s − 10·48-s + 3·49-s + 20·50-s + ⋯ |
| L(s) = 1 | − 1.41·2-s − 1.15·3-s + 3/2·4-s + 1.63·6-s + 0.755·7-s − 1.41·8-s + 1/3·9-s − 1.73·12-s − 1.06·14-s + 5/4·16-s − 0.471·18-s + 0.458·19-s − 0.872·21-s + 1.63·24-s − 2·25-s + 0.769·27-s + 1.13·28-s − 2.22·29-s − 1.06·32-s + 1/2·36-s − 0.648·38-s + 1.87·41-s + 1.23·42-s + 2.43·43-s − 1.44·48-s + 3/7·49-s + 2.82·50-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 636804 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 636804 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.5199737796\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5199737796\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 + T )^{2} \) |
| 3 | $C_2$ | \( 1 + 2 T + p T^{2} \) |
| 7 | $C_1$ | \( ( 1 - T )^{2} \) |
| 19 | $C_2$ | \( 1 - 2 T + p T^{2} \) |
| good | 5 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 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
−8.528120383242540449329483894723, −7.64939596758113403959649783851, −7.57571100088867902110310233811, −7.37702716270177429317588175316, −6.62456122444702752048713041193, −5.99620539732271804666811351997, −5.67742069637127823610337477256, −5.57928681742950427486583645839, −4.73877014588666288120041203285, −4.04211015188718094752425642434, −3.65664660676813155980150748522, −2.52986897560816951792645905521, −2.16066001255085837327128798402, −1.30325043795776141185004987249, −0.52878800471857122474052096088,
0.52878800471857122474052096088, 1.30325043795776141185004987249, 2.16066001255085837327128798402, 2.52986897560816951792645905521, 3.65664660676813155980150748522, 4.04211015188718094752425642434, 4.73877014588666288120041203285, 5.57928681742950427486583645839, 5.67742069637127823610337477256, 5.99620539732271804666811351997, 6.62456122444702752048713041193, 7.37702716270177429317588175316, 7.57571100088867902110310233811, 7.64939596758113403959649783851, 8.528120383242540449329483894723
