| L(s) = 1 | + 2-s + 3-s + 4-s − 5-s + 6-s + 7-s + 8-s + 9-s − 10-s − 11-s + 12-s − 13-s + 14-s − 15-s + 16-s + 17-s + 18-s − 19-s − 20-s + 21-s − 22-s − 23-s + 24-s + 25-s − 26-s + 27-s + 28-s + ⋯ |
| L(s) = 1 | + 2-s + 3-s + 4-s − 5-s + 6-s + 7-s + 8-s + 9-s − 10-s − 11-s + 12-s − 13-s + 14-s − 15-s + 16-s + 17-s + 18-s − 19-s − 20-s + 21-s − 22-s − 23-s + 24-s + 25-s − 26-s + 27-s + 28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 47 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 47 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(3.303548444\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.303548444\) |
| \(L(1)\) |
\(\approx\) |
\(2.291241928\) |
| \(L(1)\) |
\(\approx\) |
\(2.291241928\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 47 | \( 1 \) |
| good | 2 | \( 1 + T \) |
| 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 | \( 1 - T \) |
| 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
−33.77668607749940428936519646881, −32.05910571763781725429582287171, −31.64338490383793819126735925197, −30.61666136985830755177387698980, −29.77336479147828228394639163971, −27.91007934440203531155742388225, −26.69790885674393435467325527325, −25.43442138657100380293584968240, −24.114003617192272889938548281408, −23.58852591405474187270107134583, −21.82467104304565988628069831743, −20.76598371214385494494921133787, −19.89048518184024940264907721961, −18.64661205582266172146626732302, −16.45125695017729353171154762285, −15.06559071904897468105252816183, −14.59571799746696018494294423956, −13.054458093069987831938660166399, −11.880176109031467025983361185217, −10.40248870085373491350712870098, −8.118457993765668482753092758213, −7.40261866627296067281716249709, −5.02303396436953075466195450230, −3.77397438829552955507560998114, −2.217378213961029620175600262908,
2.217378213961029620175600262908, 3.77397438829552955507560998114, 5.02303396436953075466195450230, 7.40261866627296067281716249709, 8.118457993765668482753092758213, 10.40248870085373491350712870098, 11.880176109031467025983361185217, 13.054458093069987831938660166399, 14.59571799746696018494294423956, 15.06559071904897468105252816183, 16.45125695017729353171154762285, 18.64661205582266172146626732302, 19.89048518184024940264907721961, 20.76598371214385494494921133787, 21.82467104304565988628069831743, 23.58852591405474187270107134583, 24.114003617192272889938548281408, 25.43442138657100380293584968240, 26.69790885674393435467325527325, 27.91007934440203531155742388225, 29.77336479147828228394639163971, 30.61666136985830755177387698980, 31.64338490383793819126735925197, 32.05910571763781725429582287171, 33.77668607749940428936519646881
