| L(s) = 1 | + 2-s − 3-s + 4-s − 0.381·5-s − 6-s + 7-s + 8-s − 2·9-s − 0.381·10-s − 12-s − 5.47·13-s + 14-s + 0.381·15-s + 16-s − 1.47·17-s − 2·18-s + 6.70·19-s − 0.381·20-s − 21-s − 3.76·23-s − 24-s − 4.85·25-s − 5.47·26-s + 5·27-s + 28-s − 9.47·29-s + 0.381·30-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.577·3-s + 0.5·4-s − 0.170·5-s − 0.408·6-s + 0.377·7-s + 0.353·8-s − 0.666·9-s − 0.120·10-s − 0.288·12-s − 1.51·13-s + 0.267·14-s + 0.0986·15-s + 0.250·16-s − 0.357·17-s − 0.471·18-s + 1.53·19-s − 0.0854·20-s − 0.218·21-s − 0.784·23-s − 0.204·24-s − 0.970·25-s − 1.07·26-s + 0.962·27-s + 0.188·28-s − 1.75·29-s + 0.0697·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1694 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1694 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 - T \) |
| 7 | \( 1 - T \) |
| 11 | \( 1 \) |
| good | 3 | \( 1 + T + 3T^{2} \) |
| 5 | \( 1 + 0.381T + 5T^{2} \) |
| 13 | \( 1 + 5.47T + 13T^{2} \) |
| 17 | \( 1 + 1.47T + 17T^{2} \) |
| 19 | \( 1 - 6.70T + 19T^{2} \) |
| 23 | \( 1 + 3.76T + 23T^{2} \) |
| 29 | \( 1 + 9.47T + 29T^{2} \) |
| 31 | \( 1 + 3T + 31T^{2} \) |
| 37 | \( 1 + 1.14T + 37T^{2} \) |
| 41 | \( 1 - 6.47T + 41T^{2} \) |
| 43 | \( 1 + 9.61T + 43T^{2} \) |
| 47 | \( 1 - 1.61T + 47T^{2} \) |
| 53 | \( 1 - 0.708T + 53T^{2} \) |
| 59 | \( 1 + 5.32T + 59T^{2} \) |
| 61 | \( 1 + 13.8T + 61T^{2} \) |
| 67 | \( 1 - 4.70T + 67T^{2} \) |
| 71 | \( 1 + 3T + 71T^{2} \) |
| 73 | \( 1 + 0.145T + 73T^{2} \) |
| 79 | \( 1 + 13.9T + 79T^{2} \) |
| 83 | \( 1 - 9.32T + 83T^{2} \) |
| 89 | \( 1 - 8.09T + 89T^{2} \) |
| 97 | \( 1 - 12.4T + 97T^{2} \) |
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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
−9.061591973416945631358693144494, −7.72291806637730486976100166668, −7.48411245791431299115862190444, −6.31815898781739082870586723436, −5.48219777753432209149382497284, −5.02978252251487870865481023094, −3.98865783278496611800485833772, −2.94460435369068470172761407757, −1.86275756571368034642690696837, 0,
1.86275756571368034642690696837, 2.94460435369068470172761407757, 3.98865783278496611800485833772, 5.02978252251487870865481023094, 5.48219777753432209149382497284, 6.31815898781739082870586723436, 7.48411245791431299115862190444, 7.72291806637730486976100166668, 9.061591973416945631358693144494
