| L(s) = 1 | + 1.92·2-s + 3·3-s − 4.31·4-s − 14.0·5-s + 5.76·6-s − 13.3·7-s − 23.6·8-s + 9·9-s − 27.0·10-s − 11·11-s − 12.9·12-s + 65.5·13-s − 25.5·14-s − 42.1·15-s − 10.9·16-s + 80.9·17-s + 17.2·18-s + 83.9·19-s + 60.6·20-s − 39.9·21-s − 21.1·22-s − 41.7·23-s − 70.9·24-s + 72.6·25-s + 125.·26-s + 27·27-s + 57.4·28-s + ⋯ |
| L(s) = 1 | + 0.678·2-s + 0.577·3-s − 0.538·4-s − 1.25·5-s + 0.392·6-s − 0.719·7-s − 1.04·8-s + 0.333·9-s − 0.853·10-s − 0.301·11-s − 0.311·12-s + 1.39·13-s − 0.488·14-s − 0.726·15-s − 0.170·16-s + 1.15·17-s + 0.226·18-s + 1.01·19-s + 0.677·20-s − 0.415·21-s − 0.204·22-s − 0.378·23-s − 0.603·24-s + 0.581·25-s + 0.949·26-s + 0.192·27-s + 0.387·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2013 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2013 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 - 3T \) |
| 11 | \( 1 + 11T \) |
| 61 | \( 1 + 61T \) |
| good | 2 | \( 1 - 1.92T + 8T^{2} \) |
| 5 | \( 1 + 14.0T + 125T^{2} \) |
| 7 | \( 1 + 13.3T + 343T^{2} \) |
| 13 | \( 1 - 65.5T + 2.19e3T^{2} \) |
| 17 | \( 1 - 80.9T + 4.91e3T^{2} \) |
| 19 | \( 1 - 83.9T + 6.85e3T^{2} \) |
| 23 | \( 1 + 41.7T + 1.21e4T^{2} \) |
| 29 | \( 1 - 270.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 268.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 207.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 399.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 168.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 599.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 624.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 285.T + 2.05e5T^{2} \) |
| 67 | \( 1 - 687.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 696.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 543.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 625.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 232.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 701.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 711.T + 9.12e5T^{2} \) |
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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
−8.367007503108977953876771766152, −7.77281192519159210983879430495, −6.85291341970572515247041503433, −5.87002530143038616548597238916, −5.07040462921275376064563851131, −3.97987289209173947219990674808, −3.52631032065409110851519208211, −2.98743106972415728447728669240, −1.14334955772416832923670394221, 0,
1.14334955772416832923670394221, 2.98743106972415728447728669240, 3.52631032065409110851519208211, 3.97987289209173947219990674808, 5.07040462921275376064563851131, 5.87002530143038616548597238916, 6.85291341970572515247041503433, 7.77281192519159210983879430495, 8.367007503108977953876771766152
