| L(s) = 1 | − 2.73·2-s + 1.73·3-s + 5.46·4-s − 3.73·5-s − 4.73·6-s − 2·7-s − 9.46·8-s + 10.1·10-s + 11-s + 9.46·12-s − 3.26·13-s + 5.46·14-s − 6.46·15-s + 14.9·16-s + 17-s − 6.19·19-s − 20.3·20-s − 3.46·21-s − 2.73·22-s − 3.73·23-s − 16.3·24-s + 8.92·25-s + 8.92·26-s − 5.19·27-s − 10.9·28-s − 1.26·29-s + 17.6·30-s + ⋯ |
| L(s) = 1 | − 1.93·2-s + 1.00·3-s + 2.73·4-s − 1.66·5-s − 1.93·6-s − 0.755·7-s − 3.34·8-s + 3.22·10-s + 0.301·11-s + 2.73·12-s − 0.906·13-s + 1.46·14-s − 1.66·15-s + 3.73·16-s + 0.242·17-s − 1.42·19-s − 4.55·20-s − 0.755·21-s − 0.582·22-s − 0.778·23-s − 3.34·24-s + 1.78·25-s + 1.75·26-s − 1.00·27-s − 2.06·28-s − 0.235·29-s + 3.22·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 187 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 187 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 11 | \( 1 - T \) |
| 17 | \( 1 - T \) |
| good | 2 | \( 1 + 2.73T + 2T^{2} \) |
| 3 | \( 1 - 1.73T + 3T^{2} \) |
| 5 | \( 1 + 3.73T + 5T^{2} \) |
| 7 | \( 1 + 2T + 7T^{2} \) |
| 13 | \( 1 + 3.26T + 13T^{2} \) |
| 19 | \( 1 + 6.19T + 19T^{2} \) |
| 23 | \( 1 + 3.73T + 23T^{2} \) |
| 29 | \( 1 + 1.26T + 29T^{2} \) |
| 31 | \( 1 - 2.26T + 31T^{2} \) |
| 37 | \( 1 + 3.73T + 37T^{2} \) |
| 41 | \( 1 - 9.46T + 41T^{2} \) |
| 43 | \( 1 + 2T + 43T^{2} \) |
| 47 | \( 1 - 0.928T + 47T^{2} \) |
| 53 | \( 1 + 2.53T + 53T^{2} \) |
| 59 | \( 1 - 3T + 59T^{2} \) |
| 61 | \( 1 - 2.92T + 61T^{2} \) |
| 67 | \( 1 - T + 67T^{2} \) |
| 71 | \( 1 - 0.267T + 71T^{2} \) |
| 73 | \( 1 - 3.80T + 73T^{2} \) |
| 79 | \( 1 - 2.19T + 79T^{2} \) |
| 83 | \( 1 - 15.8T + 83T^{2} \) |
| 89 | \( 1 + 11.9T + 89T^{2} \) |
| 97 | \( 1 - 15.7T + 97T^{2} \) |
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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
−11.82544775901667069597481524111, −10.87411142962362185284498662499, −9.799025126566522876008329274843, −8.914592410369310008235797861060, −8.141491757253270606536216131037, −7.54030749408128946585418505802, −6.50059977075890955087394229485, −3.71599331035951613782649292105, −2.50911346007699238392301203647, 0,
2.50911346007699238392301203647, 3.71599331035951613782649292105, 6.50059977075890955087394229485, 7.54030749408128946585418505802, 8.141491757253270606536216131037, 8.914592410369310008235797861060, 9.799025126566522876008329274843, 10.87411142962362185284498662499, 11.82544775901667069597481524111
