| 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 + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 151 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 151 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(3.014556721\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.014556721\) |
| \(L(1)\) |
\(\approx\) |
\(1.789614290\) |
| \(L(1)\) |
\(\approx\) |
\(1.789614290\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 151 | \( 1 \) |
| 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 | \( 1 + T \) |
| 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
−28.31639243637035383327895246100, −26.8170938207008902078376664760, −25.45362243976530074383165428424, −24.78915194581082446891703634132, −23.769441551274247093244312260514, −22.554296576608170585288896585299, −22.19684626154432227977563187833, −21.38378446011155952367142902951, −20.082598518010373986500840124109, −18.9469770564855443602130152336, −17.48313884019756105728400602493, −16.70020688201458120051817777253, −15.88088563687855207996354829655, −14.45152001406646604405105210244, −13.54832954487094660878636653706, −12.37628170081708068110126599322, −11.861085498200185846416517939169, −10.25189165584762981451267850613, −9.692423216102375303828953500489, −7.276947028255453419112417065847, −6.29930926856405265256445834788, −5.624945944173553638298740527082, −4.37875642301759575700134229528, −2.86707298008006499119079026940, −1.23145043530999518656075726738,
1.23145043530999518656075726738, 2.86707298008006499119079026940, 4.37875642301759575700134229528, 5.624945944173553638298740527082, 6.29930926856405265256445834788, 7.276947028255453419112417065847, 9.692423216102375303828953500489, 10.25189165584762981451267850613, 11.861085498200185846416517939169, 12.37628170081708068110126599322, 13.54832954487094660878636653706, 14.45152001406646604405105210244, 15.88088563687855207996354829655, 16.70020688201458120051817777253, 17.48313884019756105728400602493, 18.9469770564855443602130152336, 20.082598518010373986500840124109, 21.38378446011155952367142902951, 22.19684626154432227977563187833, 22.554296576608170585288896585299, 23.769441551274247093244312260514, 24.78915194581082446891703634132, 25.45362243976530074383165428424, 26.8170938207008902078376664760, 28.31639243637035383327895246100
