| L(s) = 1 | − 275.·2-s + 7.06e3·3-s − 5.53e4·4-s − 6.20e5·5-s − 1.94e6·6-s + 2.72e7·7-s + 5.12e7·8-s − 7.92e7·9-s + 1.70e8·10-s + 6.43e7·11-s − 3.90e8·12-s + 5.86e8·13-s − 7.50e9·14-s − 4.38e9·15-s − 6.86e9·16-s − 3.10e10·17-s + 2.18e10·18-s + 5.29e10·19-s + 3.43e10·20-s + 1.92e11·21-s − 1.77e10·22-s + 1.60e11·23-s + 3.62e11·24-s − 3.77e11·25-s − 1.61e11·26-s − 1.47e12·27-s − 1.50e12·28-s + ⋯ |
| L(s) = 1 | − 0.760·2-s + 0.621·3-s − 0.422·4-s − 0.710·5-s − 0.472·6-s + 1.78·7-s + 1.08·8-s − 0.613·9-s + 0.540·10-s + 0.0905·11-s − 0.262·12-s + 0.199·13-s − 1.35·14-s − 0.441·15-s − 0.399·16-s − 1.07·17-s + 0.466·18-s + 0.715·19-s + 0.300·20-s + 1.11·21-s − 0.0688·22-s + 0.427·23-s + 0.672·24-s − 0.494·25-s − 0.151·26-s − 1.00·27-s − 0.754·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 29 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(18-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 29 ^{s/2} \, \Gamma_{\C}(s+17/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(9)\) |
\(\approx\) |
\(1.418477155\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.418477155\) |
| \(L(\frac{19}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 29 | \( 1 - 5.00e11T \) |
| good | 2 | \( 1 + 275.T + 1.31e5T^{2} \) |
| 3 | \( 1 - 7.06e3T + 1.29e8T^{2} \) |
| 5 | \( 1 + 6.20e5T + 7.62e11T^{2} \) |
| 7 | \( 1 - 2.72e7T + 2.32e14T^{2} \) |
| 11 | \( 1 - 6.43e7T + 5.05e17T^{2} \) |
| 13 | \( 1 - 5.86e8T + 8.65e18T^{2} \) |
| 17 | \( 1 + 3.10e10T + 8.27e20T^{2} \) |
| 19 | \( 1 - 5.29e10T + 5.48e21T^{2} \) |
| 23 | \( 1 - 1.60e11T + 1.41e23T^{2} \) |
| 31 | \( 1 - 7.27e11T + 2.25e25T^{2} \) |
| 37 | \( 1 + 9.77e12T + 4.56e26T^{2} \) |
| 41 | \( 1 + 6.44e13T + 2.61e27T^{2} \) |
| 43 | \( 1 - 1.42e14T + 5.87e27T^{2} \) |
| 47 | \( 1 - 1.73e14T + 2.66e28T^{2} \) |
| 53 | \( 1 + 8.85e13T + 2.05e29T^{2} \) |
| 59 | \( 1 - 4.51e14T + 1.27e30T^{2} \) |
| 61 | \( 1 - 1.64e15T + 2.24e30T^{2} \) |
| 67 | \( 1 - 7.01e14T + 1.10e31T^{2} \) |
| 71 | \( 1 - 4.44e15T + 2.96e31T^{2} \) |
| 73 | \( 1 + 8.65e14T + 4.74e31T^{2} \) |
| 79 | \( 1 - 1.17e16T + 1.81e32T^{2} \) |
| 83 | \( 1 - 1.52e16T + 4.21e32T^{2} \) |
| 89 | \( 1 - 5.11e16T + 1.37e33T^{2} \) |
| 97 | \( 1 - 7.79e16T + 5.95e33T^{2} \) |
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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
−13.63207539781406673918460970244, −11.70313907440614026503872807753, −10.81606060505932625371267981228, −9.063914360219379973363992007518, −8.310083928668023632806204507290, −7.53215352078821763274647467401, −5.10206866209399306955808362178, −3.94911336681014330019292544610, −2.07039642148117305464038240848, −0.73973602295530686199071800556,
0.73973602295530686199071800556, 2.07039642148117305464038240848, 3.94911336681014330019292544610, 5.10206866209399306955808362178, 7.53215352078821763274647467401, 8.310083928668023632806204507290, 9.063914360219379973363992007518, 10.81606060505932625371267981228, 11.70313907440614026503872807753, 13.63207539781406673918460970244
