| L(s) = 1 | − 0.197·2-s − 2.87·3-s − 1.96·4-s + 3.52·5-s + 0.566·6-s + 2.74·7-s + 0.780·8-s + 5.27·9-s − 0.695·10-s + 11-s + 5.63·12-s − 13-s − 0.540·14-s − 10.1·15-s + 3.76·16-s + 6.66·17-s − 1.03·18-s − 3.09·19-s − 6.91·20-s − 7.88·21-s − 0.197·22-s − 1.65·23-s − 2.24·24-s + 7.44·25-s + 0.197·26-s − 6.52·27-s − 5.37·28-s + ⋯ |
| L(s) = 1 | − 0.139·2-s − 1.66·3-s − 0.980·4-s + 1.57·5-s + 0.231·6-s + 1.03·7-s + 0.276·8-s + 1.75·9-s − 0.219·10-s + 0.301·11-s + 1.62·12-s − 0.277·13-s − 0.144·14-s − 2.61·15-s + 0.942·16-s + 1.61·17-s − 0.244·18-s − 0.711·19-s − 1.54·20-s − 1.72·21-s − 0.0420·22-s − 0.344·23-s − 0.458·24-s + 1.48·25-s + 0.0386·26-s − 1.25·27-s − 1.01·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 143 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 143 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.7415797997\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7415797997\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 11 | \( 1 - T \) |
| 13 | \( 1 + T \) |
| good | 2 | \( 1 + 0.197T + 2T^{2} \) |
| 3 | \( 1 + 2.87T + 3T^{2} \) |
| 5 | \( 1 - 3.52T + 5T^{2} \) |
| 7 | \( 1 - 2.74T + 7T^{2} \) |
| 17 | \( 1 - 6.66T + 17T^{2} \) |
| 19 | \( 1 + 3.09T + 19T^{2} \) |
| 23 | \( 1 + 1.65T + 23T^{2} \) |
| 29 | \( 1 + 6.61T + 29T^{2} \) |
| 31 | \( 1 - 3.31T + 31T^{2} \) |
| 37 | \( 1 - 6.26T + 37T^{2} \) |
| 41 | \( 1 - 1.08T + 41T^{2} \) |
| 43 | \( 1 - 8.39T + 43T^{2} \) |
| 47 | \( 1 + 5.08T + 47T^{2} \) |
| 53 | \( 1 + 2.48T + 53T^{2} \) |
| 59 | \( 1 + 5.30T + 59T^{2} \) |
| 61 | \( 1 + 9.01T + 61T^{2} \) |
| 67 | \( 1 + 2.57T + 67T^{2} \) |
| 71 | \( 1 + 16.6T + 71T^{2} \) |
| 73 | \( 1 - 9.22T + 73T^{2} \) |
| 79 | \( 1 - 12.4T + 79T^{2} \) |
| 83 | \( 1 + 5.00T + 83T^{2} \) |
| 89 | \( 1 + 1.35T + 89T^{2} \) |
| 97 | \( 1 + 17.1T + 97T^{2} \) |
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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
−12.97685102695069247268464834813, −12.15100228513160210766913443847, −10.95163780383433450745896038046, −10.10944670043657079511302152588, −9.341430785461436795488663550024, −7.77816046859903447841980442194, −6.10510617686103804097563780114, −5.46901883378278923414408289994, −4.55047064852309892697620320997, −1.38570195887078757213735384250,
1.38570195887078757213735384250, 4.55047064852309892697620320997, 5.46901883378278923414408289994, 6.10510617686103804097563780114, 7.77816046859903447841980442194, 9.341430785461436795488663550024, 10.10944670043657079511302152588, 10.95163780383433450745896038046, 12.15100228513160210766913443847, 12.97685102695069247268464834813
