| L(s) = 1 | + 4.67e4·3-s − 1.95e6·5-s + 3.25e7·7-s + 1.02e9·9-s − 3.59e9·11-s + 7.05e9·13-s − 9.13e10·15-s + 4.80e11·17-s − 1.17e12·19-s + 1.52e12·21-s − 5.64e12·23-s + 3.81e12·25-s − 6.42e12·27-s − 1.21e14·29-s + 1.25e14·31-s − 1.68e14·33-s − 6.36e13·35-s + 5.31e14·37-s + 3.29e14·39-s + 8.87e14·41-s − 3.63e14·43-s − 2.00e15·45-s + 8.32e15·47-s − 1.03e16·49-s + 2.24e16·51-s − 2.34e16·53-s + 7.01e15·55-s + ⋯ |
| L(s) = 1 | + 1.37·3-s − 0.447·5-s + 0.305·7-s + 0.881·9-s − 0.459·11-s + 0.184·13-s − 0.613·15-s + 0.981·17-s − 0.834·19-s + 0.418·21-s − 0.653·23-s + 0.199·25-s − 0.162·27-s − 1.54·29-s + 0.851·31-s − 0.630·33-s − 0.136·35-s + 0.672·37-s + 0.252·39-s + 0.423·41-s − 0.110·43-s − 0.394·45-s + 1.08·47-s − 0.906·49-s + 1.34·51-s − 0.977·53-s + 0.205·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 80 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(20-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 80 ^{s/2} \, \Gamma_{\C}(s+19/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(10)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{21}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 + 1.95e6T \) |
| good | 3 | \( 1 - 4.67e4T + 1.16e9T^{2} \) |
| 7 | \( 1 - 3.25e7T + 1.13e16T^{2} \) |
| 11 | \( 1 + 3.59e9T + 6.11e19T^{2} \) |
| 13 | \( 1 - 7.05e9T + 1.46e21T^{2} \) |
| 17 | \( 1 - 4.80e11T + 2.39e23T^{2} \) |
| 19 | \( 1 + 1.17e12T + 1.97e24T^{2} \) |
| 23 | \( 1 + 5.64e12T + 7.46e25T^{2} \) |
| 29 | \( 1 + 1.21e14T + 6.10e27T^{2} \) |
| 31 | \( 1 - 1.25e14T + 2.16e28T^{2} \) |
| 37 | \( 1 - 5.31e14T + 6.24e29T^{2} \) |
| 41 | \( 1 - 8.87e14T + 4.39e30T^{2} \) |
| 43 | \( 1 + 3.63e14T + 1.08e31T^{2} \) |
| 47 | \( 1 - 8.32e15T + 5.88e31T^{2} \) |
| 53 | \( 1 + 2.34e16T + 5.77e32T^{2} \) |
| 59 | \( 1 + 9.14e14T + 4.42e33T^{2} \) |
| 61 | \( 1 + 9.45e16T + 8.34e33T^{2} \) |
| 67 | \( 1 + 2.63e17T + 4.95e34T^{2} \) |
| 71 | \( 1 - 3.62e17T + 1.49e35T^{2} \) |
| 73 | \( 1 + 5.68e17T + 2.53e35T^{2} \) |
| 79 | \( 1 + 1.10e18T + 1.13e36T^{2} \) |
| 83 | \( 1 + 2.63e18T + 2.90e36T^{2} \) |
| 89 | \( 1 - 4.21e18T + 1.09e37T^{2} \) |
| 97 | \( 1 - 8.83e18T + 5.60e37T^{2} \) |
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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
−10.07800104127641572940319408900, −9.014961280887768905721125329820, −8.066267634980936090829083413133, −7.51394276309774687422603143666, −5.90765944242903166127262706331, −4.43190915859980157910089814719, −3.46900395800610637582189254618, −2.52616095608242940157219779386, −1.47491477738287589223233694723, 0,
1.47491477738287589223233694723, 2.52616095608242940157219779386, 3.46900395800610637582189254618, 4.43190915859980157910089814719, 5.90765944242903166127262706331, 7.51394276309774687422603143666, 8.066267634980936090829083413133, 9.014961280887768905721125329820, 10.07800104127641572940319408900
