| L(s) = 1 | − 0.452·2-s − 2.96·3-s − 1.79·4-s + 1.45·5-s + 1.34·6-s + 1.71·8-s + 5.79·9-s − 0.657·10-s − 11-s + 5.32·12-s + 5.76·13-s − 4.30·15-s + 2.81·16-s + 17-s − 2.62·18-s + 1.07·19-s − 2.60·20-s + 0.452·22-s + 5.87·23-s − 5.09·24-s − 2.88·25-s − 2.60·26-s − 8.28·27-s + 0.657·29-s + 1.95·30-s − 6.69·31-s − 4.70·32-s + ⋯ |
| L(s) = 1 | − 0.320·2-s − 1.71·3-s − 0.897·4-s + 0.649·5-s + 0.548·6-s + 0.607·8-s + 1.93·9-s − 0.207·10-s − 0.301·11-s + 1.53·12-s + 1.59·13-s − 1.11·15-s + 0.703·16-s + 0.242·17-s − 0.618·18-s + 0.246·19-s − 0.583·20-s + 0.0965·22-s + 1.22·23-s − 1.03·24-s − 0.577·25-s − 0.511·26-s − 1.59·27-s + 0.122·29-s + 0.356·30-s − 1.20·31-s − 0.832·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 187 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 187 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.5628731204\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5628731204\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 11 | \( 1 + T \) |
| 17 | \( 1 - T \) |
| good | 2 | \( 1 + 0.452T + 2T^{2} \) |
| 3 | \( 1 + 2.96T + 3T^{2} \) |
| 5 | \( 1 - 1.45T + 5T^{2} \) |
| 7 | \( 1 + 7T^{2} \) |
| 13 | \( 1 - 5.76T + 13T^{2} \) |
| 19 | \( 1 - 1.07T + 19T^{2} \) |
| 23 | \( 1 - 5.87T + 23T^{2} \) |
| 29 | \( 1 - 0.657T + 29T^{2} \) |
| 31 | \( 1 + 6.69T + 31T^{2} \) |
| 37 | \( 1 - 0.129T + 37T^{2} \) |
| 41 | \( 1 - 7.93T + 41T^{2} \) |
| 43 | \( 1 - 11.5T + 43T^{2} \) |
| 47 | \( 1 - 3.51T + 47T^{2} \) |
| 53 | \( 1 - 9.51T + 53T^{2} \) |
| 59 | \( 1 - 11.8T + 59T^{2} \) |
| 61 | \( 1 - 0.189T + 61T^{2} \) |
| 67 | \( 1 - 3.43T + 67T^{2} \) |
| 71 | \( 1 - 6.01T + 71T^{2} \) |
| 73 | \( 1 + 7.12T + 73T^{2} \) |
| 79 | \( 1 + 17.0T + 79T^{2} \) |
| 83 | \( 1 + 3.96T + 83T^{2} \) |
| 89 | \( 1 + 8.68T + 89T^{2} \) |
| 97 | \( 1 + 3.99T + 97T^{2} \) |
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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
−12.72218889471896105871858462652, −11.33811745809293645484979019543, −10.71359835734703909291690395909, −9.786264004508088641835832591598, −8.766878749071732736727531723808, −7.29425877017211178626997006707, −5.92925657413698345963343360435, −5.39148464802619020326167662566, −4.07199626519488671276760901683, −1.06085531184015098430651553763,
1.06085531184015098430651553763, 4.07199626519488671276760901683, 5.39148464802619020326167662566, 5.92925657413698345963343360435, 7.29425877017211178626997006707, 8.766878749071732736727531723808, 9.786264004508088641835832591598, 10.71359835734703909291690395909, 11.33811745809293645484979019543, 12.72218889471896105871858462652
