| L(s) = 1 | + 2.36·2-s + 0.569·3-s + 3.60·4-s − 4.15·5-s + 1.34·6-s − 7-s + 3.80·8-s − 2.67·9-s − 9.83·10-s + 3.47·11-s + 2.05·12-s + 1.61·13-s − 2.36·14-s − 2.36·15-s + 1.79·16-s + 17-s − 6.33·18-s + 6.35·19-s − 14.9·20-s − 0.569·21-s + 8.23·22-s + 1.21·23-s + 2.16·24-s + 12.2·25-s + 3.83·26-s − 3.23·27-s − 3.60·28-s + ⋯ |
| L(s) = 1 | + 1.67·2-s + 0.328·3-s + 1.80·4-s − 1.85·5-s + 0.550·6-s − 0.377·7-s + 1.34·8-s − 0.891·9-s − 3.11·10-s + 1.04·11-s + 0.592·12-s + 0.448·13-s − 0.632·14-s − 0.610·15-s + 0.449·16-s + 0.242·17-s − 1.49·18-s + 1.45·19-s − 3.35·20-s − 0.124·21-s + 1.75·22-s + 0.253·23-s + 0.442·24-s + 2.45·25-s + 0.751·26-s − 0.621·27-s − 0.681·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 119 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 119 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.037419744\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.037419744\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 + T \) |
| 17 | \( 1 - T \) |
| good | 2 | \( 1 - 2.36T + 2T^{2} \) |
| 3 | \( 1 - 0.569T + 3T^{2} \) |
| 5 | \( 1 + 4.15T + 5T^{2} \) |
| 11 | \( 1 - 3.47T + 11T^{2} \) |
| 13 | \( 1 - 1.61T + 13T^{2} \) |
| 19 | \( 1 - 6.35T + 19T^{2} \) |
| 23 | \( 1 - 1.21T + 23T^{2} \) |
| 29 | \( 1 + 7.97T + 29T^{2} \) |
| 31 | \( 1 + 0.939T + 31T^{2} \) |
| 37 | \( 1 + 3.61T + 37T^{2} \) |
| 41 | \( 1 + 1.30T + 41T^{2} \) |
| 43 | \( 1 + 2.52T + 43T^{2} \) |
| 47 | \( 1 + 7.87T + 47T^{2} \) |
| 53 | \( 1 - 5.67T + 53T^{2} \) |
| 59 | \( 1 + 8.70T + 59T^{2} \) |
| 61 | \( 1 - 9.30T + 61T^{2} \) |
| 67 | \( 1 + 4.89T + 67T^{2} \) |
| 71 | \( 1 - 15.0T + 71T^{2} \) |
| 73 | \( 1 - 6.78T + 73T^{2} \) |
| 79 | \( 1 - 5.59T + 79T^{2} \) |
| 83 | \( 1 - 0.756T + 83T^{2} \) |
| 89 | \( 1 - 0.619T + 89T^{2} \) |
| 97 | \( 1 - 14.9T + 97T^{2} \) |
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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
−13.59487299508271479626957510915, −12.42697968099478843260186041745, −11.65428062555370426752221510197, −11.21323329361396306912770360391, −9.067731325134918933089630268511, −7.76819068535177226189943686173, −6.69578923513706923496070020114, −5.25782545075674562549846485849, −3.78166822621131587465507209232, −3.31687337077359583248630073114,
3.31687337077359583248630073114, 3.78166822621131587465507209232, 5.25782545075674562549846485849, 6.69578923513706923496070020114, 7.76819068535177226189943686173, 9.067731325134918933089630268511, 11.21323329361396306912770360391, 11.65428062555370426752221510197, 12.42697968099478843260186041745, 13.59487299508271479626957510915
