| L(s) = 1 | + 3-s + 5-s − 7-s + 9-s + 11-s + 13-s + 15-s − 17-s + 19-s − 21-s + 23-s + 25-s + 27-s − 29-s − 31-s + 33-s − 35-s + 37-s + 39-s − 41-s − 43-s + 45-s − 47-s + 49-s − 51-s + 53-s + 55-s + ⋯ |
| L(s) = 1 | + 3-s + 5-s − 7-s + 9-s + 11-s + 13-s + 15-s − 17-s + 19-s − 21-s + 23-s + 25-s + 27-s − 29-s − 31-s + 33-s − 35-s + 37-s + 39-s − 41-s − 43-s + 45-s − 47-s + 49-s − 51-s + 53-s + 55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1336 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1336 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(2.829746002\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.829746002\) |
| \(L(1)\) |
\(\approx\) |
\(1.777225982\) |
| \(L(1)\) |
\(\approx\) |
\(1.777225982\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 167 | \( 1 \) |
| good | 3 | \( 1 + T \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 - T \) |
| 11 | \( 1 + T \) |
| 13 | \( 1 + T \) |
| 17 | \( 1 - T \) |
| 19 | \( 1 + T \) |
| 23 | \( 1 + T \) |
| 29 | \( 1 - T \) |
| 31 | \( 1 - T \) |
| 37 | \( 1 + T \) |
| 41 | \( 1 - T \) |
| 43 | \( 1 - T \) |
| 47 | \( 1 - T \) |
| 53 | \( 1 + T \) |
| 59 | \( 1 - T \) |
| 61 | \( 1 - T \) |
| 67 | \( 1 - T \) |
| 71 | \( 1 + T \) |
| 73 | \( 1 - T \) |
| 79 | \( 1 + T \) |
| 83 | \( 1 - T \) |
| 89 | \( 1 + T \) |
| 97 | \( 1 + T \) |
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\(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−20.79554358886649862415931395882, −20.08065739305387098706902632762, −19.63121233522214430071575289614, −18.42588100266243305661619492174, −18.29714683632213729806293394737, −16.94671516281537315741563123372, −16.416572073996421024766350701569, −15.44428056046160393829516234198, −14.76080230772901909000658094384, −13.87100073756023592266808380642, −13.25382149650203816802683552897, −12.91936589212092414256196232065, −11.62705360146991069512617662752, −10.63787772376808138315590004352, −9.729570027835135869598049127603, −9.1297597665002094301753830449, −8.76899520346975046678170185971, −7.38663737640540137136724127127, −6.64800831034470012615559049489, −6.00042508120669010936190753355, −4.7999784226813985589353620828, −3.63499896841451746740598450144, −3.13218734135426938834148596291, −2.00516257340494812479628918641, −1.2078869710614733825859968258,
1.2078869710614733825859968258, 2.00516257340494812479628918641, 3.13218734135426938834148596291, 3.63499896841451746740598450144, 4.7999784226813985589353620828, 6.00042508120669010936190753355, 6.64800831034470012615559049489, 7.38663737640540137136724127127, 8.76899520346975046678170185971, 9.1297597665002094301753830449, 9.729570027835135869598049127603, 10.63787772376808138315590004352, 11.62705360146991069512617662752, 12.91936589212092414256196232065, 13.25382149650203816802683552897, 13.87100073756023592266808380642, 14.76080230772901909000658094384, 15.44428056046160393829516234198, 16.416572073996421024766350701569, 16.94671516281537315741563123372, 18.29714683632213729806293394737, 18.42588100266243305661619492174, 19.63121233522214430071575289614, 20.08065739305387098706902632762, 20.79554358886649862415931395882
