| L(s) = 1 | − 0.928·2-s + 2.79·3-s − 1.13·4-s − 3.35·5-s − 2.59·6-s + 2.91·8-s + 4.82·9-s + 3.11·10-s − 0.213·11-s − 3.18·12-s + 5.00·13-s − 9.37·15-s − 0.428·16-s − 2.11·17-s − 4.47·18-s − 1.21·19-s + 3.81·20-s + 0.197·22-s − 7.54·23-s + 8.14·24-s + 6.23·25-s − 4.65·26-s + 5.10·27-s + 1.66·29-s + 8.70·30-s + 8.48·31-s − 5.42·32-s + ⋯ |
| L(s) = 1 | − 0.656·2-s + 1.61·3-s − 0.569·4-s − 1.49·5-s − 1.06·6-s + 1.03·8-s + 1.60·9-s + 0.983·10-s − 0.0642·11-s − 0.919·12-s + 1.38·13-s − 2.42·15-s − 0.107·16-s − 0.511·17-s − 1.05·18-s − 0.279·19-s + 0.852·20-s + 0.0421·22-s − 1.57·23-s + 1.66·24-s + 1.24·25-s − 0.911·26-s + 0.982·27-s + 0.309·29-s + 1.58·30-s + 1.52·31-s − 0.959·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2009 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2009 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.430908886\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.430908886\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 \) |
| 41 | \( 1 - T \) |
| good | 2 | \( 1 + 0.928T + 2T^{2} \) |
| 3 | \( 1 - 2.79T + 3T^{2} \) |
| 5 | \( 1 + 3.35T + 5T^{2} \) |
| 11 | \( 1 + 0.213T + 11T^{2} \) |
| 13 | \( 1 - 5.00T + 13T^{2} \) |
| 17 | \( 1 + 2.11T + 17T^{2} \) |
| 19 | \( 1 + 1.21T + 19T^{2} \) |
| 23 | \( 1 + 7.54T + 23T^{2} \) |
| 29 | \( 1 - 1.66T + 29T^{2} \) |
| 31 | \( 1 - 8.48T + 31T^{2} \) |
| 37 | \( 1 - 7.74T + 37T^{2} \) |
| 43 | \( 1 - 3.54T + 43T^{2} \) |
| 47 | \( 1 + 7.99T + 47T^{2} \) |
| 53 | \( 1 - 6.41T + 53T^{2} \) |
| 59 | \( 1 - 8.78T + 59T^{2} \) |
| 61 | \( 1 - 10.2T + 61T^{2} \) |
| 67 | \( 1 - 0.727T + 67T^{2} \) |
| 71 | \( 1 - 3.91T + 71T^{2} \) |
| 73 | \( 1 + 12.0T + 73T^{2} \) |
| 79 | \( 1 - 6.07T + 79T^{2} \) |
| 83 | \( 1 - 13.6T + 83T^{2} \) |
| 89 | \( 1 - 14.2T + 89T^{2} \) |
| 97 | \( 1 + 13.0T + 97T^{2} \) |
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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
−8.810407416581097234421582111758, −8.309884359886840825664335335344, −8.072355922433237567258600589537, −7.34143111877154708168741697157, −6.24773372528017863508108854410, −4.59748663880580959578266612216, −4.00001324306520306030234677821, −3.48575062341130933177908434894, −2.23583138277218655060160171042, −0.835574606554170552846985394419,
0.835574606554170552846985394419, 2.23583138277218655060160171042, 3.48575062341130933177908434894, 4.00001324306520306030234677821, 4.59748663880580959578266612216, 6.24773372528017863508108854410, 7.34143111877154708168741697157, 8.072355922433237567258600589537, 8.309884359886840825664335335344, 8.810407416581097234421582111758
