| L(s) = 1 | − 65.9·2-s + 2.25e4·3-s − 1.26e5·4-s + 3.42e5·5-s − 1.48e6·6-s + 1.63e7·7-s + 1.70e7·8-s + 3.77e8·9-s − 2.25e7·10-s + 1.38e8·11-s − 2.85e9·12-s − 3.21e9·13-s − 1.08e9·14-s + 7.70e9·15-s + 1.54e10·16-s + 3.77e10·17-s − 2.49e10·18-s − 6.09e10·19-s − 4.33e10·20-s + 3.68e11·21-s − 9.11e9·22-s + 2.42e11·23-s + 3.83e11·24-s − 6.45e11·25-s + 2.12e11·26-s + 5.59e12·27-s − 2.07e12·28-s + ⋯ |
| L(s) = 1 | − 0.182·2-s + 1.98·3-s − 0.966·4-s + 0.391·5-s − 0.361·6-s + 1.07·7-s + 0.358·8-s + 2.92·9-s − 0.0714·10-s + 0.194·11-s − 1.91·12-s − 1.09·13-s − 0.195·14-s + 0.776·15-s + 0.901·16-s + 1.31·17-s − 0.533·18-s − 0.823·19-s − 0.378·20-s + 2.12·21-s − 0.0354·22-s + 0.644·23-s + 0.710·24-s − 0.846·25-s + 0.199·26-s + 3.81·27-s − 1.03·28-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
| \(L(9)\) |
\(\approx\) |
\(4.207068883\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.207068883\) |
| \(L(\frac{19}{2})\) |
|
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 | \( 1 + 65.9T + 1.31e5T^{2} \) |
| 3 | \( 1 - 2.25e4T + 1.29e8T^{2} \) |
| 5 | \( 1 - 3.42e5T + 7.62e11T^{2} \) |
| 7 | \( 1 - 1.63e7T + 2.32e14T^{2} \) |
| 11 | \( 1 - 1.38e8T + 5.05e17T^{2} \) |
| 13 | \( 1 + 3.21e9T + 8.65e18T^{2} \) |
| 17 | \( 1 - 3.77e10T + 8.27e20T^{2} \) |
| 19 | \( 1 + 6.09e10T + 5.48e21T^{2} \) |
| 23 | \( 1 - 2.42e11T + 1.41e23T^{2} \) |
| 31 | \( 1 + 1.83e12T + 2.25e25T^{2} \) |
| 37 | \( 1 - 1.75e13T + 4.56e26T^{2} \) |
| 41 | \( 1 - 1.19e13T + 2.61e27T^{2} \) |
| 43 | \( 1 + 5.09e13T + 5.87e27T^{2} \) |
| 47 | \( 1 - 5.99e13T + 2.66e28T^{2} \) |
| 53 | \( 1 - 8.34e14T + 2.05e29T^{2} \) |
| 59 | \( 1 + 9.87e14T + 1.27e30T^{2} \) |
| 61 | \( 1 + 5.79e14T + 2.24e30T^{2} \) |
| 67 | \( 1 - 5.24e15T + 1.10e31T^{2} \) |
| 71 | \( 1 - 9.25e15T + 2.96e31T^{2} \) |
| 73 | \( 1 + 1.03e16T + 4.74e31T^{2} \) |
| 79 | \( 1 - 1.94e16T + 1.81e32T^{2} \) |
| 83 | \( 1 + 1.07e16T + 4.21e32T^{2} \) |
| 89 | \( 1 - 5.92e15T + 1.37e33T^{2} \) |
| 97 | \( 1 - 7.11e16T + 5.95e33T^{2} \) |
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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
−13.65111526819581776161960890359, −12.53555306715005891446065904265, −10.15280096882885778688988875025, −9.326264109196336308353509678790, −8.285071617364294842997621973962, −7.52136660134845884699759650780, −4.91721648647419635251342337678, −3.79321950580185850083484152900, −2.33469169549888160245326462961, −1.21406171056893342959581573897,
1.21406171056893342959581573897, 2.33469169549888160245326462961, 3.79321950580185850083484152900, 4.91721648647419635251342337678, 7.52136660134845884699759650780, 8.285071617364294842997621973962, 9.326264109196336308353509678790, 10.15280096882885778688988875025, 12.53555306715005891446065904265, 13.65111526819581776161960890359
