| L(s) = 1 | − 3-s − 5-s + 3·9-s + 8·11-s − 13-s + 15-s − 3·17-s + 8·19-s − 5·23-s + 5·25-s − 8·27-s + 7·29-s + 8·31-s − 8·33-s − 20·37-s + 39-s + 5·41-s − 5·43-s − 3·45-s + 7·47-s − 14·49-s + 3·51-s + 11·53-s − 8·55-s − 8·57-s + 3·59-s + 11·61-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.447·5-s + 9-s + 2.41·11-s − 0.277·13-s + 0.258·15-s − 0.727·17-s + 1.83·19-s − 1.04·23-s + 25-s − 1.53·27-s + 1.29·29-s + 1.43·31-s − 1.39·33-s − 3.28·37-s + 0.160·39-s + 0.780·41-s − 0.762·43-s − 0.447·45-s + 1.02·47-s − 2·49-s + 0.420·51-s + 1.51·53-s − 1.07·55-s − 1.05·57-s + 0.390·59-s + 1.40·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1478656 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1478656 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.038716461\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.038716461\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | | \( 1 \) |
| 19 | $C_2$ | \( 1 - 8 T + p T^{2} \) |
| good | 3 | $C_2^2$ | \( 1 + T - 2 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 5 | $C_2^2$ | \( 1 + T - 4 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 7 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 + T - 12 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 + 3 T - 8 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 + 5 T + 2 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 7 T + 20 T^{2} - 7 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 - 5 T - 16 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 13 T + p T^{2} ) \) |
| 47 | $C_2^2$ | \( 1 - 7 T + 2 T^{2} - 7 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 11 T + 68 T^{2} - 11 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 3 T - 50 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 11 T + 60 T^{2} - 11 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 + 3 T - 58 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2^2$ | \( 1 + 11 T + 50 T^{2} + 11 p T^{3} + p^{2} T^{4} \) |
| 73 | $C_2^2$ | \( 1 + 15 T + 152 T^{2} + 15 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 - 17 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 + 3 T - 80 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 - 19 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
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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
−10.12974116851784177510739424791, −9.535410791987473980052116502366, −9.149145462644269915291853308443, −8.730656008192374881332195779733, −8.441077738281850344115536236223, −7.82648993991234874058193208123, −7.32789131671831894718956048886, −6.87775854573007508450641997933, −6.73413549737044777708227204023, −6.39919526618954127540283321712, −5.69132326049427070035321836526, −5.30244719325971240695223039780, −4.72791856684776711902182837348, −4.26937653309723728699824070656, −3.96214577637191392787099964113, −3.42643847339027405608035657755, −2.89103846145515756479984791114, −1.83657513146358670977229837723, −1.41569620587877553977564110850, −0.70474159750067721955262695646,
0.70474159750067721955262695646, 1.41569620587877553977564110850, 1.83657513146358670977229837723, 2.89103846145515756479984791114, 3.42643847339027405608035657755, 3.96214577637191392787099964113, 4.26937653309723728699824070656, 4.72791856684776711902182837348, 5.30244719325971240695223039780, 5.69132326049427070035321836526, 6.39919526618954127540283321712, 6.73413549737044777708227204023, 6.87775854573007508450641997933, 7.32789131671831894718956048886, 7.82648993991234874058193208123, 8.441077738281850344115536236223, 8.730656008192374881332195779733, 9.149145462644269915291853308443, 9.535410791987473980052116502366, 10.12974116851784177510739424791
