| L(s) = 1 | − 1.79·2-s − 0.497·3-s + 1.23·4-s − 2.09·5-s + 0.894·6-s + 2.50·7-s + 1.37·8-s − 2.75·9-s + 3.76·10-s − 5.64·11-s − 0.613·12-s + 5.23·13-s − 4.49·14-s + 1.04·15-s − 4.94·16-s − 17-s + 4.95·18-s + 1.48·19-s − 2.58·20-s − 1.24·21-s + 10.1·22-s − 1.59·23-s − 0.684·24-s − 0.617·25-s − 9.42·26-s + 2.86·27-s + 3.08·28-s + ⋯ |
| L(s) = 1 | − 1.27·2-s − 0.287·3-s + 0.617·4-s − 0.936·5-s + 0.365·6-s + 0.945·7-s + 0.486·8-s − 0.917·9-s + 1.19·10-s − 1.70·11-s − 0.177·12-s + 1.45·13-s − 1.20·14-s + 0.268·15-s − 1.23·16-s − 0.242·17-s + 1.16·18-s + 0.339·19-s − 0.577·20-s − 0.271·21-s + 2.16·22-s − 0.332·23-s − 0.139·24-s − 0.123·25-s − 1.84·26-s + 0.550·27-s + 0.583·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 731 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 731 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.4638665404\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4638665404\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 17 | \( 1 + T \) |
| 43 | \( 1 - T \) |
| good | 2 | \( 1 + 1.79T + 2T^{2} \) |
| 3 | \( 1 + 0.497T + 3T^{2} \) |
| 5 | \( 1 + 2.09T + 5T^{2} \) |
| 7 | \( 1 - 2.50T + 7T^{2} \) |
| 11 | \( 1 + 5.64T + 11T^{2} \) |
| 13 | \( 1 - 5.23T + 13T^{2} \) |
| 19 | \( 1 - 1.48T + 19T^{2} \) |
| 23 | \( 1 + 1.59T + 23T^{2} \) |
| 29 | \( 1 + 2.50T + 29T^{2} \) |
| 31 | \( 1 + 4.16T + 31T^{2} \) |
| 37 | \( 1 - 7.99T + 37T^{2} \) |
| 41 | \( 1 - 11.7T + 41T^{2} \) |
| 47 | \( 1 - 9.21T + 47T^{2} \) |
| 53 | \( 1 + 5.98T + 53T^{2} \) |
| 59 | \( 1 - 6.65T + 59T^{2} \) |
| 61 | \( 1 - 3.15T + 61T^{2} \) |
| 67 | \( 1 - 12.4T + 67T^{2} \) |
| 71 | \( 1 + 0.0484T + 71T^{2} \) |
| 73 | \( 1 - 8.47T + 73T^{2} \) |
| 79 | \( 1 + 3.03T + 79T^{2} \) |
| 83 | \( 1 - 13.6T + 83T^{2} \) |
| 89 | \( 1 - 8.23T + 89T^{2} \) |
| 97 | \( 1 + 16.3T + 97T^{2} \) |
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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
−10.58912133774326365369406057116, −9.375992635113385352026650388644, −8.462536664638327699999879894385, −7.972130697013282336916880014764, −7.50214925551280092775440577819, −5.98846291647391662915656370118, −5.03774973772480428576513209564, −3.88981136233428390254430861216, −2.34079388918854983641814751479, −0.68659342818445091999445293713,
0.68659342818445091999445293713, 2.34079388918854983641814751479, 3.88981136233428390254430861216, 5.03774973772480428576513209564, 5.98846291647391662915656370118, 7.50214925551280092775440577819, 7.972130697013282336916880014764, 8.462536664638327699999879894385, 9.375992635113385352026650388644, 10.58912133774326365369406057116
