| L(s) = 1 | − 2·7-s − 3·13-s + 2·19-s + 25-s − 43-s + 49-s − 61-s − 3·67-s + 73-s + 3·79-s + 6·91-s − 2·121-s + 127-s + 131-s − 4·133-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 5·169-s + 173-s − 2·175-s + 179-s + 181-s + ⋯ |
| L(s) = 1 | − 2·7-s − 3·13-s + 2·19-s + 25-s − 43-s + 49-s − 61-s − 3·67-s + 73-s + 3·79-s + 6·91-s − 2·121-s + 127-s + 131-s − 4·133-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 5·169-s + 173-s − 2·175-s + 179-s + 181-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7485696 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7485696 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(0.5529633901\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5529633901\) |
| \(L(1)\) |
|
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 | | \( 1 \) |
| 3 | | \( 1 \) |
| 19 | $C_1$ | \( ( 1 - T )^{2} \) |
| good | 5 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 7 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 13 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{2}( 1 + T + T^{2} ) \) |
| 17 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 23 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 29 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 43 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{2}( 1 - T + T^{2} ) \) |
| 47 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 53 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 61 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{2}( 1 - T + T^{2} ) \) |
| 67 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{2}( 1 + T + T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 73 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 + T + T^{2} ) \) |
| 79 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 - T + T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 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
−9.441745388027253689956561372730, −8.995895180678542776277115577399, −8.537485418341376265690210624144, −7.81681491718672206284192146434, −7.53189804481116670919883617897, −7.43020527753381586090306463394, −6.91681211065318024499585023462, −6.52321417514903310232637302742, −6.34826758260039717661456176861, −5.71516146990152639360945575936, −5.17710922816259131839983574731, −4.99497728007132371328790400023, −4.67648375550538696526344388330, −3.96229600867309390172410672302, −3.38553863517960802531946724868, −3.01845737590230745737867451544, −2.83421409205258426244237504865, −2.29035475215703533085928477681, −1.49569383600972657544422535565, −0.46855008322279034826699165570,
0.46855008322279034826699165570, 1.49569383600972657544422535565, 2.29035475215703533085928477681, 2.83421409205258426244237504865, 3.01845737590230745737867451544, 3.38553863517960802531946724868, 3.96229600867309390172410672302, 4.67648375550538696526344388330, 4.99497728007132371328790400023, 5.17710922816259131839983574731, 5.71516146990152639360945575936, 6.34826758260039717661456176861, 6.52321417514903310232637302742, 6.91681211065318024499585023462, 7.43020527753381586090306463394, 7.53189804481116670919883617897, 7.81681491718672206284192146434, 8.537485418341376265690210624144, 8.995895180678542776277115577399, 9.441745388027253689956561372730
