| L(s) = 1 | + 2·3-s − 4-s − 2·7-s + 3·9-s − 2·12-s − 2·13-s + 19-s − 4·21-s − 25-s + 4·27-s + 2·28-s + 31-s − 3·36-s − 2·37-s − 4·39-s − 2·43-s + 49-s + 2·52-s + 2·57-s + 61-s − 6·63-s + 64-s + 67-s + 2·73-s − 2·75-s − 76-s + 79-s + ⋯ |
| L(s) = 1 | + 2·3-s − 4-s − 2·7-s + 3·9-s − 2·12-s − 2·13-s + 19-s − 4·21-s − 25-s + 4·27-s + 2·28-s + 31-s − 3·36-s − 2·37-s − 4·39-s − 2·43-s + 49-s + 2·52-s + 2·57-s + 61-s − 6·63-s + 64-s + 67-s + 2·73-s − 2·75-s − 76-s + 79-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 47961 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 47961 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(0.6340336221\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6340336221\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | $C_1$ | \( ( 1 - T )^{2} \) |
| 73 | $C_1$ | \( ( 1 - T )^{2} \) |
| good | 2 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 5 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 7 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 17 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 19 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 + T + T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 31 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 + T + T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 61 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 + T + T^{2} ) \) |
| 67 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 + T + T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 79 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 + T + T^{2} ) \) |
| 83 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 89 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
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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
−12.74719977044241681720134263674, −12.65108401578661403128764928241, −12.12062743888975156313483944535, −11.50322858806305280593848097340, −10.11911989962710840090273180383, −10.11654712722514798768635364969, −9.789127368282544569976937339697, −9.411058356722090518296638297526, −8.986172789151246223874999450802, −8.440833091900710946942655077511, −7.88471468849690736423507932513, −7.36530332073772196555495306206, −6.77622737527666176902204422573, −6.49933111326190171752140952962, −5.03874340334624992464181583430, −4.94663228189786512255944111978, −3.78269200159433631376921749222, −3.56043158964368782384867652424, −2.84876783781709283044854490220, −2.14024990608216911688125917658,
2.14024990608216911688125917658, 2.84876783781709283044854490220, 3.56043158964368782384867652424, 3.78269200159433631376921749222, 4.94663228189786512255944111978, 5.03874340334624992464181583430, 6.49933111326190171752140952962, 6.77622737527666176902204422573, 7.36530332073772196555495306206, 7.88471468849690736423507932513, 8.440833091900710946942655077511, 8.986172789151246223874999450802, 9.411058356722090518296638297526, 9.789127368282544569976937339697, 10.11654712722514798768635364969, 10.11911989962710840090273180383, 11.50322858806305280593848097340, 12.12062743888975156313483944535, 12.65108401578661403128764928241, 12.74719977044241681720134263674
