| L(s) = 1 | − 0.749·2-s + 0.436·3-s − 1.43·4-s − 3.66·5-s − 0.327·6-s − 2.83·7-s + 2.57·8-s − 2.80·9-s + 2.75·10-s − 11-s − 0.628·12-s − 6.70·13-s + 2.12·14-s − 1.60·15-s + 0.944·16-s − 7.82·17-s + 2.10·18-s − 1.04·19-s + 5.27·20-s − 1.23·21-s + 0.749·22-s + 3.68·23-s + 1.12·24-s + 8.46·25-s + 5.02·26-s − 2.53·27-s + 4.08·28-s + ⋯ |
| L(s) = 1 | − 0.530·2-s + 0.252·3-s − 0.719·4-s − 1.64·5-s − 0.133·6-s − 1.07·7-s + 0.911·8-s − 0.936·9-s + 0.869·10-s − 0.301·11-s − 0.181·12-s − 1.85·13-s + 0.568·14-s − 0.413·15-s + 0.236·16-s − 1.89·17-s + 0.496·18-s − 0.239·19-s + 1.17·20-s − 0.270·21-s + 0.159·22-s + 0.768·23-s + 0.229·24-s + 1.69·25-s + 0.985·26-s − 0.488·27-s + 0.771·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1441 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.01515940425\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.01515940425\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 11 | \( 1 + T \) |
| 131 | \( 1 - T \) |
| good | 2 | \( 1 + 0.749T + 2T^{2} \) |
| 3 | \( 1 - 0.436T + 3T^{2} \) |
| 5 | \( 1 + 3.66T + 5T^{2} \) |
| 7 | \( 1 + 2.83T + 7T^{2} \) |
| 13 | \( 1 + 6.70T + 13T^{2} \) |
| 17 | \( 1 + 7.82T + 17T^{2} \) |
| 19 | \( 1 + 1.04T + 19T^{2} \) |
| 23 | \( 1 - 3.68T + 23T^{2} \) |
| 29 | \( 1 + 1.52T + 29T^{2} \) |
| 31 | \( 1 - 1.24T + 31T^{2} \) |
| 37 | \( 1 - 2.23T + 37T^{2} \) |
| 41 | \( 1 - 2.70T + 41T^{2} \) |
| 43 | \( 1 + 9.75T + 43T^{2} \) |
| 47 | \( 1 + 7.21T + 47T^{2} \) |
| 53 | \( 1 + 2.07T + 53T^{2} \) |
| 59 | \( 1 + 5.31T + 59T^{2} \) |
| 61 | \( 1 - 5.83T + 61T^{2} \) |
| 67 | \( 1 + 7.17T + 67T^{2} \) |
| 71 | \( 1 - 11.8T + 71T^{2} \) |
| 73 | \( 1 - 7.01T + 73T^{2} \) |
| 79 | \( 1 + 10.1T + 79T^{2} \) |
| 83 | \( 1 + 15.5T + 83T^{2} \) |
| 89 | \( 1 + 1.07T + 89T^{2} \) |
| 97 | \( 1 - 0.738T + 97T^{2} \) |
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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
−9.361173844323056176337339377058, −8.717190246656603588547793217186, −8.060951818391919493432727125806, −7.31838486069174192524998161570, −6.58820085870831911616539720693, −5.05539283556902994465559998059, −4.45366622700811167752649029002, −3.44980359985662169149254631043, −2.55284059103007598521782771343, −0.087243166460801231701202976065,
0.087243166460801231701202976065, 2.55284059103007598521782771343, 3.44980359985662169149254631043, 4.45366622700811167752649029002, 5.05539283556902994465559998059, 6.58820085870831911616539720693, 7.31838486069174192524998161570, 8.060951818391919493432727125806, 8.717190246656603588547793217186, 9.361173844323056176337339377058
