| L(s) = 1 | + 1.81·2-s + 1.20·3-s + 1.28·4-s − 5-s + 2.18·6-s − 1.28·8-s − 1.54·9-s − 1.81·10-s − 11-s + 1.55·12-s − 4.21·13-s − 1.20·15-s − 4.91·16-s + 1.71·17-s − 2.80·18-s − 2.74·19-s − 1.28·20-s − 1.81·22-s + 4.93·23-s − 1.55·24-s + 25-s − 7.65·26-s − 5.48·27-s + 2.93·29-s − 2.18·30-s − 2.51·31-s − 6.34·32-s + ⋯ |
| L(s) = 1 | + 1.28·2-s + 0.696·3-s + 0.644·4-s − 0.447·5-s + 0.892·6-s − 0.455·8-s − 0.515·9-s − 0.573·10-s − 0.301·11-s + 0.448·12-s − 1.17·13-s − 0.311·15-s − 1.22·16-s + 0.415·17-s − 0.660·18-s − 0.629·19-s − 0.288·20-s − 0.386·22-s + 1.02·23-s − 0.317·24-s + 0.200·25-s − 1.50·26-s − 1.05·27-s + 0.545·29-s − 0.399·30-s − 0.451·31-s − 1.12·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2695 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2695 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + T \) |
| 7 | \( 1 \) |
| 11 | \( 1 + T \) |
| good | 2 | \( 1 - 1.81T + 2T^{2} \) |
| 3 | \( 1 - 1.20T + 3T^{2} \) |
| 13 | \( 1 + 4.21T + 13T^{2} \) |
| 17 | \( 1 - 1.71T + 17T^{2} \) |
| 19 | \( 1 + 2.74T + 19T^{2} \) |
| 23 | \( 1 - 4.93T + 23T^{2} \) |
| 29 | \( 1 - 2.93T + 29T^{2} \) |
| 31 | \( 1 + 2.51T + 31T^{2} \) |
| 37 | \( 1 - 5.81T + 37T^{2} \) |
| 41 | \( 1 + 8.57T + 41T^{2} \) |
| 43 | \( 1 + 7.38T + 43T^{2} \) |
| 47 | \( 1 + 11.3T + 47T^{2} \) |
| 53 | \( 1 + 12.1T + 53T^{2} \) |
| 59 | \( 1 + 3.07T + 59T^{2} \) |
| 61 | \( 1 + 8.10T + 61T^{2} \) |
| 67 | \( 1 - 1.81T + 67T^{2} \) |
| 71 | \( 1 - 1.33T + 71T^{2} \) |
| 73 | \( 1 + 9.74T + 73T^{2} \) |
| 79 | \( 1 - 12.2T + 79T^{2} \) |
| 83 | \( 1 - 8.75T + 83T^{2} \) |
| 89 | \( 1 - 6.24T + 89T^{2} \) |
| 97 | \( 1 - 1.61T + 97T^{2} \) |
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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
−8.363180329143843083296932667142, −7.71571082701745229150781244208, −6.79967698173967512816628311739, −6.03126383587532202710686766096, −4.95139893432997383877050755323, −4.70796623418537936023747844244, −3.39849113588686722432291315713, −3.08723939327633545479733782422, −2.08329649789993858756052081627, 0,
2.08329649789993858756052081627, 3.08723939327633545479733782422, 3.39849113588686722432291315713, 4.70796623418537936023747844244, 4.95139893432997383877050755323, 6.03126383587532202710686766096, 6.79967698173967512816628311739, 7.71571082701745229150781244208, 8.363180329143843083296932667142
