| L(s) = 1 | + 0.488·2-s + 2.60·3-s − 1.76·4-s + 1.92·5-s + 1.27·6-s + 0.259·7-s − 1.83·8-s + 3.78·9-s + 0.940·10-s + 11-s − 4.58·12-s − 2.14·13-s + 0.126·14-s + 5.02·15-s + 2.62·16-s + 4.30·17-s + 1.84·18-s + 7.44·19-s − 3.39·20-s + 0.675·21-s + 0.488·22-s + 3.82·23-s − 4.78·24-s − 1.28·25-s − 1.04·26-s + 2.05·27-s − 0.456·28-s + ⋯ |
| L(s) = 1 | + 0.345·2-s + 1.50·3-s − 0.880·4-s + 0.861·5-s + 0.519·6-s + 0.0979·7-s − 0.649·8-s + 1.26·9-s + 0.297·10-s + 0.301·11-s − 1.32·12-s − 0.596·13-s + 0.0338·14-s + 1.29·15-s + 0.656·16-s + 1.04·17-s + 0.435·18-s + 1.70·19-s − 0.759·20-s + 0.147·21-s + 0.104·22-s + 0.798·23-s − 0.976·24-s − 0.257·25-s − 0.205·26-s + 0.394·27-s − 0.0862·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 671 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 671 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.760171235\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.760171235\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 11 | \( 1 - T \) |
| 61 | \( 1 + T \) |
| good | 2 | \( 1 - 0.488T + 2T^{2} \) |
| 3 | \( 1 - 2.60T + 3T^{2} \) |
| 5 | \( 1 - 1.92T + 5T^{2} \) |
| 7 | \( 1 - 0.259T + 7T^{2} \) |
| 13 | \( 1 + 2.14T + 13T^{2} \) |
| 17 | \( 1 - 4.30T + 17T^{2} \) |
| 19 | \( 1 - 7.44T + 19T^{2} \) |
| 23 | \( 1 - 3.82T + 23T^{2} \) |
| 29 | \( 1 + 3.58T + 29T^{2} \) |
| 31 | \( 1 + 8.02T + 31T^{2} \) |
| 37 | \( 1 + 6.82T + 37T^{2} \) |
| 41 | \( 1 - 5.88T + 41T^{2} \) |
| 43 | \( 1 - 7.55T + 43T^{2} \) |
| 47 | \( 1 + 13.1T + 47T^{2} \) |
| 53 | \( 1 + 10.1T + 53T^{2} \) |
| 59 | \( 1 - 5.99T + 59T^{2} \) |
| 67 | \( 1 + 1.73T + 67T^{2} \) |
| 71 | \( 1 - 11.3T + 71T^{2} \) |
| 73 | \( 1 + 1.24T + 73T^{2} \) |
| 79 | \( 1 + 9.01T + 79T^{2} \) |
| 83 | \( 1 + 13.5T + 83T^{2} \) |
| 89 | \( 1 - 5.65T + 89T^{2} \) |
| 97 | \( 1 + 17.0T + 97T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.910526152811014225803239682825, −9.552482900898166536563911557255, −9.008908616800739864852825953376, −7.971677324435485710669027950212, −7.27545646423812355946384664369, −5.73979691351212991063819337663, −5.00712174269454988157128365991, −3.66331421344895814785981932254, −3.01018280837511490738763373049, −1.59366754730560254070291747796,
1.59366754730560254070291747796, 3.01018280837511490738763373049, 3.66331421344895814785981932254, 5.00712174269454988157128365991, 5.73979691351212991063819337663, 7.27545646423812355946384664369, 7.971677324435485710669027950212, 9.008908616800739864852825953376, 9.552482900898166536563911557255, 9.910526152811014225803239682825
