| L(s) = 1 | − 2·2-s − 3-s + 3·4-s + 2·6-s − 4·8-s + 3·11-s − 3·12-s + 5·16-s − 3·17-s + 19-s − 6·22-s + 4·24-s + 25-s + 27-s − 6·32-s − 3·33-s + 6·34-s − 2·38-s − 2·41-s + 2·43-s + 9·44-s − 5·48-s − 49-s − 2·50-s + 3·51-s − 2·54-s − 57-s + ⋯ |
| L(s) = 1 | − 2·2-s − 3-s + 3·4-s + 2·6-s − 4·8-s + 3·11-s − 3·12-s + 5·16-s − 3·17-s + 19-s − 6·22-s + 4·24-s + 25-s + 27-s − 6·32-s − 3·33-s + 6·34-s − 2·38-s − 2·41-s + 2·43-s + 9·44-s − 5·48-s − 49-s − 2·50-s + 3·51-s − 2·54-s − 57-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1871424 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1871424 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(0.3410043853\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3410043853\) |
| \(L(1)\) |
|
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 | $C_1$ | \( ( 1 + T )^{2} \) |
| 3 | $C_2$ | \( 1 + T + T^{2} \) |
| 19 | $C_2$ | \( 1 - T + T^{2} \) |
| good | 5 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 7 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 11 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 - T + T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 17 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{2}( 1 + T + T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 31 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 37 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - T + T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 53 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 - T + T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 67 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 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 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 83 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 - T + T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 - T + T^{2} )^{2} \) |
| 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
−9.721355463959134691161741118030, −9.610367479125074542375809165438, −8.969248259023872714596308549739, −8.860147512753636417823836705957, −8.752769495195393819796676600604, −8.206131462325004199271734023771, −7.36220772233599304417868007460, −7.18135705888495785062953160495, −6.58723751399258118460112075738, −6.58003573195336483460411007912, −6.23629704951720967258355848582, −5.81754995497184410893113815522, −4.93378302681367424376007331369, −4.66502369776871945739365329030, −3.64975772518846747556985400865, −3.59988044271356916795739855126, −2.55606483507367598836196841208, −2.08902770502707213204060737146, −1.37413255242098916729067087745, −0.793951325935148882853601502898,
0.793951325935148882853601502898, 1.37413255242098916729067087745, 2.08902770502707213204060737146, 2.55606483507367598836196841208, 3.59988044271356916795739855126, 3.64975772518846747556985400865, 4.66502369776871945739365329030, 4.93378302681367424376007331369, 5.81754995497184410893113815522, 6.23629704951720967258355848582, 6.58003573195336483460411007912, 6.58723751399258118460112075738, 7.18135705888495785062953160495, 7.36220772233599304417868007460, 8.206131462325004199271734023771, 8.752769495195393819796676600604, 8.860147512753636417823836705957, 8.969248259023872714596308549739, 9.610367479125074542375809165438, 9.721355463959134691161741118030
