| L(s) = 1 | + 2·2-s − 4-s + 4·7-s − 8·8-s + 9-s + 2·11-s + 8·14-s − 7·16-s + 2·18-s + 4·22-s + 16·23-s − 6·25-s − 4·28-s − 12·29-s + 14·32-s − 36-s + 12·37-s − 2·44-s + 32·46-s + 9·49-s − 12·50-s + 12·53-s − 32·56-s − 24·58-s + 4·63-s + 35·64-s − 8·67-s + ⋯ |
| L(s) = 1 | + 1.41·2-s − 1/2·4-s + 1.51·7-s − 2.82·8-s + 1/3·9-s + 0.603·11-s + 2.13·14-s − 7/4·16-s + 0.471·18-s + 0.852·22-s + 3.33·23-s − 6/5·25-s − 0.755·28-s − 2.22·29-s + 2.47·32-s − 1/6·36-s + 1.97·37-s − 0.301·44-s + 4.71·46-s + 9/7·49-s − 1.69·50-s + 1.64·53-s − 4.27·56-s − 3.15·58-s + 0.503·63-s + 35/8·64-s − 0.977·67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 53361 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 53361 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.325810380\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.325810380\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 - T )^{2} \) |
| good | 2 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) |
| 5 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 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 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 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 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 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
−9.861943929631485923831880412642, −9.430206051937599176170301208447, −8.975261543279013968212899152873, −8.732088684533910814314758308014, −7.933480025037517782290916340443, −7.46995816184461358823952634116, −6.78773519865467313405876423609, −5.99045197964575513149352902495, −5.31622482563829752699303340190, −5.25930136350710548023686434400, −4.35338404351630392777693329438, −4.20407903283329927801590362766, −3.44264814031822825656923586814, −2.58943983398263715091283271039, −1.20030448276572071441538926359,
1.20030448276572071441538926359, 2.58943983398263715091283271039, 3.44264814031822825656923586814, 4.20407903283329927801590362766, 4.35338404351630392777693329438, 5.25930136350710548023686434400, 5.31622482563829752699303340190, 5.99045197964575513149352902495, 6.78773519865467313405876423609, 7.46995816184461358823952634116, 7.933480025037517782290916340443, 8.732088684533910814314758308014, 8.975261543279013968212899152873, 9.430206051937599176170301208447, 9.861943929631485923831880412642
