| L(s) = 1 | − 0.0673·2-s − 8.14·3-s − 7.99·4-s − 19.7·5-s + 0.548·6-s − 21.7·7-s + 1.07·8-s + 39.3·9-s + 1.32·10-s − 48.7·11-s + 65.1·12-s − 4.06·13-s + 1.46·14-s + 160.·15-s + 63.8·16-s + 17·17-s − 2.65·18-s + 0.449·19-s + 157.·20-s + 176.·21-s + 3.28·22-s − 181.·23-s − 8.77·24-s + 264.·25-s + 0.273·26-s − 100.·27-s + 173.·28-s + ⋯ |
| L(s) = 1 | − 0.0238·2-s − 1.56·3-s − 0.999·4-s − 1.76·5-s + 0.0373·6-s − 1.17·7-s + 0.0475·8-s + 1.45·9-s + 0.0420·10-s − 1.33·11-s + 1.56·12-s − 0.0867·13-s + 0.0279·14-s + 2.76·15-s + 0.998·16-s + 0.242·17-s − 0.0347·18-s + 0.00542·19-s + 1.76·20-s + 1.83·21-s + 0.0318·22-s − 1.64·23-s − 0.0746·24-s + 2.11·25-s + 0.00206·26-s − 0.718·27-s + 1.17·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 799 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 799 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 17 | \( 1 - 17T \) |
| 47 | \( 1 + 47T \) |
| good | 2 | \( 1 + 0.0673T + 8T^{2} \) |
| 3 | \( 1 + 8.14T + 27T^{2} \) |
| 5 | \( 1 + 19.7T + 125T^{2} \) |
| 7 | \( 1 + 21.7T + 343T^{2} \) |
| 11 | \( 1 + 48.7T + 1.33e3T^{2} \) |
| 13 | \( 1 + 4.06T + 2.19e3T^{2} \) |
| 19 | \( 1 - 0.449T + 6.85e3T^{2} \) |
| 23 | \( 1 + 181.T + 1.21e4T^{2} \) |
| 29 | \( 1 + 28.2T + 2.43e4T^{2} \) |
| 31 | \( 1 - 179.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 212.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 492.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 160.T + 7.95e4T^{2} \) |
| 53 | \( 1 + 617.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 613.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 213.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 589.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 788.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 59.2T + 3.89e5T^{2} \) |
| 79 | \( 1 + 978.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 35.6T + 5.71e5T^{2} \) |
| 89 | \( 1 + 801.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 707.T + 9.12e5T^{2} \) |
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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
−9.825578248153752158390151605117, −8.353012342795233069580288505022, −7.80346301262384475606977511415, −6.76250821681794302161943207604, −5.80126676110185454621827041021, −4.90309343442806820589940297956, −4.15358101051372590886345883421, −3.20307057925005261829076825489, −0.56984424944317047597598984404, 0,
0.56984424944317047597598984404, 3.20307057925005261829076825489, 4.15358101051372590886345883421, 4.90309343442806820589940297956, 5.80126676110185454621827041021, 6.76250821681794302161943207604, 7.80346301262384475606977511415, 8.353012342795233069580288505022, 9.825578248153752158390151605117
