| L(s) = 1 | − 109.·2-s − 2.08e4·4-s + 9.35e4·5-s + 3.01e6·7-s + 5.85e6·8-s − 1.02e7·10-s − 8.72e7·11-s + 5.76e7·13-s − 3.29e8·14-s + 4.25e7·16-s + 1.85e9·17-s − 6.88e9·19-s − 1.94e9·20-s + 9.52e9·22-s + 6.99e9·23-s − 2.17e10·25-s − 6.30e9·26-s − 6.28e10·28-s + 9.15e10·29-s − 1.80e10·31-s − 1.96e11·32-s − 2.02e11·34-s + 2.82e11·35-s + 6.30e11·37-s + 7.51e11·38-s + 5.48e11·40-s + 1.88e12·41-s + ⋯ |
| L(s) = 1 | − 0.603·2-s − 0.635·4-s + 0.535·5-s + 1.38·7-s + 0.987·8-s − 0.323·10-s − 1.34·11-s + 0.254·13-s − 0.835·14-s + 0.0396·16-s + 1.09·17-s − 1.76·19-s − 0.340·20-s + 0.814·22-s + 0.428·23-s − 0.712·25-s − 0.153·26-s − 0.880·28-s + 0.985·29-s − 0.117·31-s − 1.01·32-s − 0.661·34-s + 0.741·35-s + 1.09·37-s + 1.06·38-s + 0.528·40-s + 1.51·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 27 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(16-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 27 ^{s/2} \, \Gamma_{\C}(s+15/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(8)\) |
\(\approx\) |
\(1.469428210\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.469428210\) |
| \(L(\frac{17}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| good | 2 | \( 1 + 109.T + 3.27e4T^{2} \) |
| 5 | \( 1 - 9.35e4T + 3.05e10T^{2} \) |
| 7 | \( 1 - 3.01e6T + 4.74e12T^{2} \) |
| 11 | \( 1 + 8.72e7T + 4.17e15T^{2} \) |
| 13 | \( 1 - 5.76e7T + 5.11e16T^{2} \) |
| 17 | \( 1 - 1.85e9T + 2.86e18T^{2} \) |
| 19 | \( 1 + 6.88e9T + 1.51e19T^{2} \) |
| 23 | \( 1 - 6.99e9T + 2.66e20T^{2} \) |
| 29 | \( 1 - 9.15e10T + 8.62e21T^{2} \) |
| 31 | \( 1 + 1.80e10T + 2.34e22T^{2} \) |
| 37 | \( 1 - 6.30e11T + 3.33e23T^{2} \) |
| 41 | \( 1 - 1.88e12T + 1.55e24T^{2} \) |
| 43 | \( 1 - 4.43e10T + 3.17e24T^{2} \) |
| 47 | \( 1 + 5.73e11T + 1.20e25T^{2} \) |
| 53 | \( 1 - 1.48e13T + 7.31e25T^{2} \) |
| 59 | \( 1 - 1.50e13T + 3.65e26T^{2} \) |
| 61 | \( 1 - 8.84e12T + 6.02e26T^{2} \) |
| 67 | \( 1 + 7.14e13T + 2.46e27T^{2} \) |
| 71 | \( 1 - 9.50e13T + 5.87e27T^{2} \) |
| 73 | \( 1 - 8.27e13T + 8.90e27T^{2} \) |
| 79 | \( 1 - 1.40e14T + 2.91e28T^{2} \) |
| 83 | \( 1 + 1.74e14T + 6.11e28T^{2} \) |
| 89 | \( 1 + 2.92e14T + 1.74e29T^{2} \) |
| 97 | \( 1 + 2.86e14T + 6.33e29T^{2} \) |
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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
−13.89529753857431555797727897143, −12.78825795694228485727033876937, −10.96467188731420606295880934990, −10.04712333071217033767436355148, −8.531892662200743966980320852410, −7.75947050160305411811371164605, −5.56504393480026178253811214675, −4.42529491648020568476262711181, −2.17717628762618457368751163653, −0.820112497667995978975077592180,
0.820112497667995978975077592180, 2.17717628762618457368751163653, 4.42529491648020568476262711181, 5.56504393480026178253811214675, 7.75947050160305411811371164605, 8.531892662200743966980320852410, 10.04712333071217033767436355148, 10.96467188731420606295880934990, 12.78825795694228485727033876937, 13.89529753857431555797727897143
