| L(s) = 1 | + 2.42·2-s + 1.58·3-s + 3.89·4-s + 3.57·5-s + 3.83·6-s − 0.709·7-s + 4.59·8-s − 0.498·9-s + 8.68·10-s − 11-s + 6.15·12-s − 4.54·13-s − 1.72·14-s + 5.66·15-s + 3.37·16-s − 1.39·17-s − 1.21·18-s − 3.83·19-s + 13.9·20-s − 1.12·21-s − 2.42·22-s + 7.53·23-s + 7.26·24-s + 7.80·25-s − 11.0·26-s − 5.53·27-s − 2.76·28-s + ⋯ |
| L(s) = 1 | + 1.71·2-s + 0.913·3-s + 1.94·4-s + 1.60·5-s + 1.56·6-s − 0.268·7-s + 1.62·8-s − 0.166·9-s + 2.74·10-s − 0.301·11-s + 1.77·12-s − 1.25·13-s − 0.460·14-s + 1.46·15-s + 0.842·16-s − 0.337·17-s − 0.285·18-s − 0.879·19-s + 3.11·20-s − 0.245·21-s − 0.517·22-s + 1.57·23-s + 1.48·24-s + 1.56·25-s − 2.16·26-s − 1.06·27-s − 0.522·28-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
| \(L(1)\) |
\(\approx\) |
\(6.861143985\) |
| \(L(\frac12)\) |
\(\approx\) |
\(6.861143985\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - 2.42T + 2T^{2} \) |
| 3 | \( 1 - 1.58T + 3T^{2} \) |
| 5 | \( 1 - 3.57T + 5T^{2} \) |
| 7 | \( 1 + 0.709T + 7T^{2} \) |
| 13 | \( 1 + 4.54T + 13T^{2} \) |
| 17 | \( 1 + 1.39T + 17T^{2} \) |
| 19 | \( 1 + 3.83T + 19T^{2} \) |
| 23 | \( 1 - 7.53T + 23T^{2} \) |
| 29 | \( 1 - 0.952T + 29T^{2} \) |
| 31 | \( 1 + 1.74T + 31T^{2} \) |
| 37 | \( 1 - 0.496T + 37T^{2} \) |
| 41 | \( 1 + 2.02T + 41T^{2} \) |
| 43 | \( 1 - 8.85T + 43T^{2} \) |
| 47 | \( 1 + 1.33T + 47T^{2} \) |
| 53 | \( 1 - 10.8T + 53T^{2} \) |
| 59 | \( 1 + 13.4T + 59T^{2} \) |
| 61 | \( 1 + 13.4T + 61T^{2} \) |
| 67 | \( 1 - 9.96T + 67T^{2} \) |
| 71 | \( 1 + 9.05T + 71T^{2} \) |
| 73 | \( 1 - 13.1T + 73T^{2} \) |
| 79 | \( 1 - 14.9T + 79T^{2} \) |
| 83 | \( 1 + 5.92T + 83T^{2} \) |
| 89 | \( 1 - 15.5T + 89T^{2} \) |
| 97 | \( 1 - 12.1T + 97T^{2} \) |
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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
−9.407912403586294301158944848208, −8.973441671062737796512199864795, −7.65187204709738861646449254239, −6.74835374595415688920115098812, −6.06372313065037510811519456401, −5.25160105580352529290818138391, −4.60952539504004144486615406721, −3.29673889323598402695387719020, −2.56693714022205210934534920182, −2.04138931132644409497883557701,
2.04138931132644409497883557701, 2.56693714022205210934534920182, 3.29673889323598402695387719020, 4.60952539504004144486615406721, 5.25160105580352529290818138391, 6.06372313065037510811519456401, 6.74835374595415688920115098812, 7.65187204709738861646449254239, 8.973441671062737796512199864795, 9.407912403586294301158944848208
