| L(s) = 1 | − 1.89·2-s + 2.98·3-s + 1.58·4-s + 3.71·5-s − 5.65·6-s + 2.24·7-s + 0.780·8-s + 5.89·9-s − 7.04·10-s − 11-s + 4.73·12-s + 1.33·13-s − 4.26·14-s + 11.0·15-s − 4.65·16-s − 0.829·17-s − 11.1·18-s − 2.17·19-s + 5.90·20-s + 6.71·21-s + 1.89·22-s + 2.63·23-s + 2.32·24-s + 8.81·25-s − 2.52·26-s + 8.65·27-s + 3.57·28-s + ⋯ |
| L(s) = 1 | − 1.33·2-s + 1.72·3-s + 0.794·4-s + 1.66·5-s − 2.30·6-s + 0.850·7-s + 0.275·8-s + 1.96·9-s − 2.22·10-s − 0.301·11-s + 1.36·12-s + 0.370·13-s − 1.13·14-s + 2.86·15-s − 1.16·16-s − 0.201·17-s − 2.63·18-s − 0.499·19-s + 1.32·20-s + 1.46·21-s + 0.403·22-s + 0.549·23-s + 0.474·24-s + 1.76·25-s − 0.496·26-s + 1.66·27-s + 0.675·28-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
| \(L(1)\) |
\(\approx\) |
\(2.296932470\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.296932470\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + 1.89T + 2T^{2} \) |
| 3 | \( 1 - 2.98T + 3T^{2} \) |
| 5 | \( 1 - 3.71T + 5T^{2} \) |
| 7 | \( 1 - 2.24T + 7T^{2} \) |
| 13 | \( 1 - 1.33T + 13T^{2} \) |
| 17 | \( 1 + 0.829T + 17T^{2} \) |
| 19 | \( 1 + 2.17T + 19T^{2} \) |
| 23 | \( 1 - 2.63T + 23T^{2} \) |
| 29 | \( 1 + 6.18T + 29T^{2} \) |
| 31 | \( 1 + 1.56T + 31T^{2} \) |
| 37 | \( 1 - 2.72T + 37T^{2} \) |
| 41 | \( 1 + 7.63T + 41T^{2} \) |
| 43 | \( 1 + 3.49T + 43T^{2} \) |
| 47 | \( 1 + 6.02T + 47T^{2} \) |
| 53 | \( 1 - 7.36T + 53T^{2} \) |
| 59 | \( 1 - 6.02T + 59T^{2} \) |
| 61 | \( 1 + 3.28T + 61T^{2} \) |
| 67 | \( 1 - 1.56T + 67T^{2} \) |
| 71 | \( 1 + 12.2T + 71T^{2} \) |
| 73 | \( 1 + 13.6T + 73T^{2} \) |
| 79 | \( 1 + 11.4T + 79T^{2} \) |
| 83 | \( 1 - 9.75T + 83T^{2} \) |
| 89 | \( 1 + 8.59T + 89T^{2} \) |
| 97 | \( 1 - 9.51T + 97T^{2} \) |
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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
−9.341450444644775301703847182569, −8.764716939631938596816629731567, −8.321532242284532438422011511690, −7.47556808689848717351893498695, −6.69568503210271223724465462072, −5.41921993069946477403167714380, −4.34637304224375506025876831937, −2.93234398620502909363964588419, −1.93265718691719207628831890815, −1.55240854025923176604853210425,
1.55240854025923176604853210425, 1.93265718691719207628831890815, 2.93234398620502909363964588419, 4.34637304224375506025876831937, 5.41921993069946477403167714380, 6.69568503210271223724465462072, 7.47556808689848717351893498695, 8.321532242284532438422011511690, 8.764716939631938596816629731567, 9.341450444644775301703847182569
