L(s) = 1 | − 3·2-s − 3-s + 4·4-s − 6·5-s + 3·6-s − 7-s − 3·8-s + 18·10-s − 4·12-s − 9·13-s + 3·14-s + 6·15-s + 3·16-s − 6·17-s − 6·19-s − 24·20-s + 21-s + 3·24-s + 19·25-s + 27·26-s + 27-s − 4·28-s − 18·30-s − 6·32-s + 18·34-s + 6·35-s − 10·37-s + ⋯ |
L(s) = 1 | − 2.12·2-s − 0.577·3-s + 2·4-s − 2.68·5-s + 1.22·6-s − 0.377·7-s − 1.06·8-s + 5.69·10-s − 1.15·12-s − 2.49·13-s + 0.801·14-s + 1.54·15-s + 3/4·16-s − 1.45·17-s − 1.37·19-s − 5.36·20-s + 0.218·21-s + 0.612·24-s + 19/5·25-s + 5.29·26-s + 0.192·27-s − 0.755·28-s − 3.28·30-s − 1.06·32-s + 3.08·34-s + 1.01·35-s − 1.64·37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 12321 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 12321 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
---|
bad | 3 | $C_2$ | \( 1 + T + T^{2} \) |
| 37 | $C_2$ | \( 1 + 10 T + p T^{2} \) |
good | 2 | $C_2^2$ | \( 1 + 3 T + 5 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 5 | $C_2^2$ | \( 1 + 6 T + 17 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 7 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 11 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 + 2 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 17 | $C_2^2$ | \( 1 + 6 T + 29 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 23 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2^2$ | \( 1 - 35 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2^2$ | \( 1 - 6 T - 5 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 - 59 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 53 | $C_2^2$ | \( 1 + 12 T + 91 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 24 T + 251 T^{2} - 24 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + p T^{2} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 + 7 T - 18 T^{2} + 7 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2^2$ | \( 1 - 6 T - 35 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 73 | $C_2$ | \( ( 1 - 7 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 13 T + p T^{2} ) \) |
| 83 | $C_2^2$ | \( 1 - 6 T - 47 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 + 24 T + 281 T^{2} + 24 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 + 49 T^{2} + p^{2} T^{4} \) |
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
−12.94988809746636458183299566952, −12.48434739870057257824566414875, −12.21165918698668434162996881177, −11.66306559702870213093798131226, −11.03713929211594532081223599661, −10.92195502142850016346224136573, −9.987262794008242628987798647283, −9.775020382879372427649050473806, −8.824506457122171371991179665324, −8.599597614948697678685665348376, −7.985551136281917579492170905665, −7.58790311086488824377797535584, −6.94887604808944865242968641125, −6.67900542768103785569722484279, −5.16534014516323065010330018291, −4.47852998814285297512193731646, −3.79606960382670372045779972142, −2.52759037219994674746512217344, 0, 0,
2.52759037219994674746512217344, 3.79606960382670372045779972142, 4.47852998814285297512193731646, 5.16534014516323065010330018291, 6.67900542768103785569722484279, 6.94887604808944865242968641125, 7.58790311086488824377797535584, 7.985551136281917579492170905665, 8.599597614948697678685665348376, 8.824506457122171371991179665324, 9.775020382879372427649050473806, 9.987262794008242628987798647283, 10.92195502142850016346224136573, 11.03713929211594532081223599661, 11.66306559702870213093798131226, 12.21165918698668434162996881177, 12.48434739870057257824566414875, 12.94988809746636458183299566952