| L(s) = 1 | − 3·4-s − 4·7-s + 9-s + 11-s + 4·13-s + 5·16-s + 4·17-s + 16·23-s − 6·25-s + 12·28-s − 3·36-s + 12·37-s + 4·41-s − 3·44-s + 9·49-s − 12·52-s + 12·53-s − 12·61-s − 4·63-s − 3·64-s − 8·67-s − 12·68-s + 28·73-s − 4·77-s + 81-s − 24·83-s − 16·91-s + ⋯ |
| L(s) = 1 | − 3/2·4-s − 1.51·7-s + 1/3·9-s + 0.301·11-s + 1.10·13-s + 5/4·16-s + 0.970·17-s + 3.33·23-s − 6/5·25-s + 2.26·28-s − 1/2·36-s + 1.97·37-s + 0.624·41-s − 0.452·44-s + 9/7·49-s − 1.66·52-s + 1.64·53-s − 1.53·61-s − 0.503·63-s − 3/8·64-s − 0.977·67-s − 1.45·68-s + 3.27·73-s − 0.455·77-s + 1/9·81-s − 2.63·83-s − 1.67·91-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 586971 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 586971 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.287699630\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.287699630\) |
| \(L(\frac{3}{2})\) |
|
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_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| 7 | $C_2$ | \( 1 + 4 T + p T^{2} \) |
| 11 | $C_1$ | \( 1 - T \) |
| good | 2 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 5 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 14 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.732088684533910814314758308014, −7.933480025037517782290916340443, −7.67376126887882242787787369865, −6.85553577597705614902881023761, −6.78773519865467313405876423609, −5.99045197964575513149352902495, −5.66464594246847012421851082372, −5.25930136350710548023686434400, −4.35338404351630392777693329438, −4.33065348032451946577135661956, −3.44264814031822825656923586814, −3.31119773648628104035349562784, −2.58943983398263715091283271039, −1.20030448276572071441538926359, −0.71891279628917143719551815370,
0.71891279628917143719551815370, 1.20030448276572071441538926359, 2.58943983398263715091283271039, 3.31119773648628104035349562784, 3.44264814031822825656923586814, 4.33065348032451946577135661956, 4.35338404351630392777693329438, 5.25930136350710548023686434400, 5.66464594246847012421851082372, 5.99045197964575513149352902495, 6.78773519865467313405876423609, 6.85553577597705614902881023761, 7.67376126887882242787787369865, 7.933480025037517782290916340443, 8.732088684533910814314758308014
