| L(s) = 1 | − 1.91e6·2-s − 1.04e10·3-s − 5.12e12·4-s + 2.05e15·5-s + 2.00e16·6-s + 1.37e18·7-s + 2.66e19·8-s + 1.09e20·9-s − 3.94e21·10-s − 2.95e22·11-s + 5.36e22·12-s + 4.85e23·13-s − 2.62e24·14-s − 2.15e25·15-s − 6.01e24·16-s + 2.08e26·17-s − 2.09e26·18-s − 1.07e27·19-s − 1.05e28·20-s − 1.43e28·21-s + 5.66e28·22-s − 2.47e29·23-s − 2.79e29·24-s + 3.10e30·25-s − 9.29e29·26-s − 1.14e30·27-s − 7.02e30·28-s + ⋯ |
| L(s) = 1 | − 0.645·2-s − 0.577·3-s − 0.582·4-s + 1.93·5-s + 0.372·6-s + 0.927·7-s + 1.02·8-s + 0.333·9-s − 1.24·10-s − 1.20·11-s + 0.336·12-s + 0.544·13-s − 0.599·14-s − 1.11·15-s − 0.0777·16-s + 0.730·17-s − 0.215·18-s − 0.346·19-s − 1.12·20-s − 0.535·21-s + 0.778·22-s − 1.30·23-s − 0.590·24-s + 2.72·25-s − 0.351·26-s − 0.192·27-s − 0.540·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(44-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3 ^{s/2} \, \Gamma_{\C}(s+43/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(22)\) |
\(\approx\) |
\(1.587353910\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.587353910\) |
| \(L(\frac{45}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + 1.04e10T \) |
| good | 2 | \( 1 + 1.91e6T + 8.79e12T^{2} \) |
| 5 | \( 1 - 2.05e15T + 1.13e30T^{2} \) |
| 7 | \( 1 - 1.37e18T + 2.18e36T^{2} \) |
| 11 | \( 1 + 2.95e22T + 6.02e44T^{2} \) |
| 13 | \( 1 - 4.85e23T + 7.93e47T^{2} \) |
| 17 | \( 1 - 2.08e26T + 8.11e52T^{2} \) |
| 19 | \( 1 + 1.07e27T + 9.69e54T^{2} \) |
| 23 | \( 1 + 2.47e29T + 3.58e58T^{2} \) |
| 29 | \( 1 - 1.39e31T + 7.64e62T^{2} \) |
| 31 | \( 1 - 1.48e31T + 1.34e64T^{2} \) |
| 37 | \( 1 - 9.02e33T + 2.70e67T^{2} \) |
| 41 | \( 1 - 5.58e34T + 2.23e69T^{2} \) |
| 43 | \( 1 + 2.06e35T + 1.73e70T^{2} \) |
| 47 | \( 1 - 6.46e35T + 7.94e71T^{2} \) |
| 53 | \( 1 + 4.21e36T + 1.39e74T^{2} \) |
| 59 | \( 1 - 4.40e37T + 1.40e76T^{2} \) |
| 61 | \( 1 - 4.53e37T + 5.87e76T^{2} \) |
| 67 | \( 1 + 1.64e37T + 3.32e78T^{2} \) |
| 71 | \( 1 - 6.75e39T + 4.01e79T^{2} \) |
| 73 | \( 1 + 5.28e39T + 1.32e80T^{2} \) |
| 79 | \( 1 - 1.01e41T + 3.96e81T^{2} \) |
| 83 | \( 1 - 2.11e41T + 3.31e82T^{2} \) |
| 89 | \( 1 + 7.34e41T + 6.66e83T^{2} \) |
| 97 | \( 1 + 6.37e42T + 2.69e85T^{2} \) |
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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
−16.67786549549510485042935214862, −14.16206999297844609742296785551, −13.06584550600153549327846829325, −10.64482458264040886278501707103, −9.730561944451866828097338665167, −8.110784709379787080397612230849, −5.90984680020422613384031993088, −4.86721887913704089652919089642, −2.06743632948506199627412055768, −0.927450751436717633333564789955,
0.927450751436717633333564789955, 2.06743632948506199627412055768, 4.86721887913704089652919089642, 5.90984680020422613384031993088, 8.110784709379787080397612230849, 9.730561944451866828097338665167, 10.64482458264040886278501707103, 13.06584550600153549327846829325, 14.16206999297844609742296785551, 16.67786549549510485042935214862
