| L(s) = 1 | + 2-s + 2·3-s + 4-s + 2·6-s + 8-s + 9-s + 2·12-s − 8·13-s + 16-s + 18-s + 2·24-s − 10·25-s − 8·26-s − 4·27-s + 32-s + 36-s + 4·37-s − 16·39-s + 24·47-s + 2·48-s + 49-s − 10·50-s − 8·52-s − 4·54-s + 12·59-s + 16·61-s + 64-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1.15·3-s + 1/2·4-s + 0.816·6-s + 0.353·8-s + 1/3·9-s + 0.577·12-s − 2.21·13-s + 1/4·16-s + 0.235·18-s + 0.408·24-s − 2·25-s − 1.56·26-s − 0.769·27-s + 0.176·32-s + 1/6·36-s + 0.657·37-s − 2.56·39-s + 3.50·47-s + 0.288·48-s + 1/7·49-s − 1.41·50-s − 1.10·52-s − 0.544·54-s + 1.56·59-s + 2.04·61-s + 1/8·64-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 14112 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 14112 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.028724583\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.028724583\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( 1 - T \) |
| 3 | $C_2$ | \( 1 - 2 T + p T^{2} \) |
| 7 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| good | 5 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 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 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 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
−11.52629384186915375545261187858, −10.47027853370113893616936981615, −10.11465006559287644470603673937, −9.392494157732895222790142944171, −9.207333736191082013174831123730, −8.182643116839158569055135937831, −7.80365099617056766338064266882, −7.29668061464756137953129595214, −6.71310127772875365703449938682, −5.62168100149868099368912779374, −5.31921059379060528260176389250, −4.19102102405251164216473944262, −3.82418745638366191622617161116, −2.48514181668809031419660323366, −2.39101862347891857041372548490,
2.39101862347891857041372548490, 2.48514181668809031419660323366, 3.82418745638366191622617161116, 4.19102102405251164216473944262, 5.31921059379060528260176389250, 5.62168100149868099368912779374, 6.71310127772875365703449938682, 7.29668061464756137953129595214, 7.80365099617056766338064266882, 8.182643116839158569055135937831, 9.207333736191082013174831123730, 9.392494157732895222790142944171, 10.11465006559287644470603673937, 10.47027853370113893616936981615, 11.52629384186915375545261187858
