| 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 + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 167 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 167 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(4.866354931\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.866354931\) |
| \(L(1)\) |
\(\approx\) |
\(2.674141120\) |
| \(L(1)\) |
\(\approx\) |
\(2.674141120\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 167 | \( 1 \) |
| 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
−27.12378997696251126932343629156, −26.55067729087980361596688488588, −25.01839483409371023341255140590, −24.44335021401190102510577246563, −23.79156592748344192310437534829, −22.39511193068526158470617753319, −21.675926712234030550814060695810, −20.3332484821427846576579691981, −20.000812550287204443936068508584, −19.02351871502123810502864806765, −17.44753296711892310000970463333, −15.989985310313805925833198019420, −15.21197989110277424128471557917, −14.39791096305423865934616191341, −13.69948282833158366412064852097, −12.165020232413149780335181510111, −11.680588866232009791626147215397, −10.24333318938611204163358042818, −8.6361062743218147830895494653, −7.660304846122071627529116616997, −6.78623094240610713082252751499, −4.81582690846231672896616581630, −4.12799351167152547436340224824, −2.91493591240312607412751784655, −1.567014621378652972449943036,
1.567014621378652972449943036, 2.91493591240312607412751784655, 4.12799351167152547436340224824, 4.81582690846231672896616581630, 6.78623094240610713082252751499, 7.660304846122071627529116616997, 8.6361062743218147830895494653, 10.24333318938611204163358042818, 11.680588866232009791626147215397, 12.165020232413149780335181510111, 13.69948282833158366412064852097, 14.39791096305423865934616191341, 15.21197989110277424128471557917, 15.989985310313805925833198019420, 17.44753296711892310000970463333, 19.02351871502123810502864806765, 20.000812550287204443936068508584, 20.3332484821427846576579691981, 21.675926712234030550814060695810, 22.39511193068526158470617753319, 23.79156592748344192310437534829, 24.44335021401190102510577246563, 25.01839483409371023341255140590, 26.55067729087980361596688488588, 27.12378997696251126932343629156
