| L(s) = 1 | − 4.44·2-s + 6.83·3-s + 11.7·4-s − 2.20·5-s − 30.3·6-s − 16.6·8-s + 19.7·9-s + 9.78·10-s − 3.73·11-s + 80.3·12-s + 89.7·13-s − 15.0·15-s − 19.9·16-s − 55.9·17-s − 87.7·18-s − 27.6·19-s − 25.8·20-s + 16.5·22-s + 55.7·23-s − 113.·24-s − 120.·25-s − 399.·26-s − 49.6·27-s + 99.5·29-s + 66.9·30-s + 99.2·31-s + 221.·32-s + ⋯ |
| L(s) = 1 | − 1.57·2-s + 1.31·3-s + 1.46·4-s − 0.197·5-s − 2.06·6-s − 0.736·8-s + 0.731·9-s + 0.309·10-s − 0.102·11-s + 1.93·12-s + 1.91·13-s − 0.259·15-s − 0.311·16-s − 0.797·17-s − 1.14·18-s − 0.333·19-s − 0.289·20-s + 0.160·22-s + 0.505·23-s − 0.968·24-s − 0.961·25-s − 3.00·26-s − 0.353·27-s + 0.637·29-s + 0.407·30-s + 0.574·31-s + 1.22·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2401 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2401 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.603599259\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.603599259\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 \) |
| good | 2 | \( 1 + 4.44T + 8T^{2} \) |
| 3 | \( 1 - 6.83T + 27T^{2} \) |
| 5 | \( 1 + 2.20T + 125T^{2} \) |
| 11 | \( 1 + 3.73T + 1.33e3T^{2} \) |
| 13 | \( 1 - 89.7T + 2.19e3T^{2} \) |
| 17 | \( 1 + 55.9T + 4.91e3T^{2} \) |
| 19 | \( 1 + 27.6T + 6.85e3T^{2} \) |
| 23 | \( 1 - 55.7T + 1.21e4T^{2} \) |
| 29 | \( 1 - 99.5T + 2.43e4T^{2} \) |
| 31 | \( 1 - 99.2T + 2.97e4T^{2} \) |
| 37 | \( 1 + 358.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 333.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 425.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 483.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 407.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 572.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 153.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 370.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 386.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 678.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 677.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 792.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 128.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 430.T + 9.12e5T^{2} \) |
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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
−8.616512005035609057314730105508, −8.320402570203998404916010354662, −7.39733758247203217362816080359, −6.75062389761171468238048997892, −5.78611968650550359211921621717, −4.27251823502031537844897117018, −3.51341528997628780165333394615, −2.47775509484751594687972543512, −1.69697922421580957982373888572, −0.68036359587697146130755381217,
0.68036359587697146130755381217, 1.69697922421580957982373888572, 2.47775509484751594687972543512, 3.51341528997628780165333394615, 4.27251823502031537844897117018, 5.78611968650550359211921621717, 6.75062389761171468238048997892, 7.39733758247203217362816080359, 8.320402570203998404916010354662, 8.616512005035609057314730105508
