| L(s) = 1 | − 2-s + 1.14·3-s + 4-s − 0.375·5-s − 1.14·6-s + 0.176·7-s − 8-s − 1.69·9-s + 0.375·10-s + 11-s + 1.14·12-s + 5.77·13-s − 0.176·14-s − 0.427·15-s + 16-s − 1.61·17-s + 1.69·18-s + 5.31·19-s − 0.375·20-s + 0.201·21-s − 22-s + 6.15·23-s − 1.14·24-s − 4.85·25-s − 5.77·26-s − 5.35·27-s + 0.176·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.658·3-s + 0.5·4-s − 0.167·5-s − 0.465·6-s + 0.0668·7-s − 0.353·8-s − 0.566·9-s + 0.118·10-s + 0.301·11-s + 0.329·12-s + 1.60·13-s − 0.0472·14-s − 0.110·15-s + 0.250·16-s − 0.390·17-s + 0.400·18-s + 1.21·19-s − 0.0838·20-s + 0.0440·21-s − 0.213·22-s + 1.28·23-s − 0.232·24-s − 0.971·25-s − 1.13·26-s − 1.03·27-s + 0.0334·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4334 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4334 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.819699768\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.819699768\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 11 | \( 1 - T \) |
| 197 | \( 1 + T \) |
| good | 3 | \( 1 - 1.14T + 3T^{2} \) |
| 5 | \( 1 + 0.375T + 5T^{2} \) |
| 7 | \( 1 - 0.176T + 7T^{2} \) |
| 13 | \( 1 - 5.77T + 13T^{2} \) |
| 17 | \( 1 + 1.61T + 17T^{2} \) |
| 19 | \( 1 - 5.31T + 19T^{2} \) |
| 23 | \( 1 - 6.15T + 23T^{2} \) |
| 29 | \( 1 - 3.06T + 29T^{2} \) |
| 31 | \( 1 - 8.87T + 31T^{2} \) |
| 37 | \( 1 + 3.49T + 37T^{2} \) |
| 41 | \( 1 + 11.5T + 41T^{2} \) |
| 43 | \( 1 - 8.97T + 43T^{2} \) |
| 47 | \( 1 - 7.36T + 47T^{2} \) |
| 53 | \( 1 + 11.0T + 53T^{2} \) |
| 59 | \( 1 + 10.0T + 59T^{2} \) |
| 61 | \( 1 - 1.83T + 61T^{2} \) |
| 67 | \( 1 - 1.50T + 67T^{2} \) |
| 71 | \( 1 - 8.55T + 71T^{2} \) |
| 73 | \( 1 + 6.63T + 73T^{2} \) |
| 79 | \( 1 - 2.84T + 79T^{2} \) |
| 83 | \( 1 - 7.86T + 83T^{2} \) |
| 89 | \( 1 - 0.885T + 89T^{2} \) |
| 97 | \( 1 - 7.17T + 97T^{2} \) |
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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
−8.276953703896406340774952285561, −8.030137422692514739195284928070, −7.02893875650609353485760493511, −6.32614458811036961031398750358, −5.60473081932904421347436420440, −4.54640080708887107010508820595, −3.41393136110978638267137608132, −3.03853838510587547496048903470, −1.81995985207056283719626691344, −0.848645576766222137101499992389,
0.848645576766222137101499992389, 1.81995985207056283719626691344, 3.03853838510587547496048903470, 3.41393136110978638267137608132, 4.54640080708887107010508820595, 5.60473081932904421347436420440, 6.32614458811036961031398750358, 7.02893875650609353485760493511, 8.030137422692514739195284928070, 8.276953703896406340774952285561
