| L(s) = 1 | − 2-s + 4-s + 5-s − 7-s − 8-s − 10-s + 11-s + 13-s + 14-s + 16-s + 19-s + 20-s − 22-s + 23-s + 25-s − 26-s − 28-s + 29-s − 31-s − 32-s − 35-s − 37-s − 38-s − 40-s + 41-s + 43-s + 44-s + ⋯ |
| L(s) = 1 | − 2-s + 4-s + 5-s − 7-s − 8-s − 10-s + 11-s + 13-s + 14-s + 16-s + 19-s + 20-s − 22-s + 23-s + 25-s − 26-s − 28-s + 29-s − 31-s − 32-s − 35-s − 37-s − 38-s − 40-s + 41-s + 43-s + 44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 51 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 51 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(1.170343006\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.170343006\) |
| \(L(1)\) |
\(\approx\) |
\(0.8798219249\) |
| \(L(1)\) |
\(\approx\) |
\(0.8798219249\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 17 | \( 1 \) |
| good | 2 | \( 1 - T \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 - T \) |
| 11 | \( 1 + T \) |
| 13 | \( 1 + T \) |
| 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
−33.09869394216564081159271016679, −32.79324138370125404493943352195, −30.70184816406532019097135596160, −29.46215763381464506773198725632, −28.817917596257666582009131918744, −27.640015665735691382734448525715, −26.26235870190654948660914894829, −25.45301265391250462208825059694, −24.59344331055706125032484083793, −22.76267450500585636101443204126, −21.44366500133620954157291946194, −20.23558873162871550875155429587, −19.06464386143708039029066352161, −17.92663682454653922236327297602, −16.83154612982430641502771913646, −15.804262608644272073404801132213, −14.0996495675154905864822826984, −12.60915778160047693382425905940, −11.00335155908981631145070351494, −9.68157968562557281317528926504, −8.90344823883406652361740796753, −6.92450645055312231767231855101, −5.92634090948389307180183727785, −3.13894144062052336872300193917, −1.26789064670733730699447432749,
1.26789064670733730699447432749, 3.13894144062052336872300193917, 5.92634090948389307180183727785, 6.92450645055312231767231855101, 8.90344823883406652361740796753, 9.68157968562557281317528926504, 11.00335155908981631145070351494, 12.60915778160047693382425905940, 14.0996495675154905864822826984, 15.804262608644272073404801132213, 16.83154612982430641502771913646, 17.92663682454653922236327297602, 19.06464386143708039029066352161, 20.23558873162871550875155429587, 21.44366500133620954157291946194, 22.76267450500585636101443204126, 24.59344331055706125032484083793, 25.45301265391250462208825059694, 26.26235870190654948660914894829, 27.640015665735691382734448525715, 28.817917596257666582009131918744, 29.46215763381464506773198725632, 30.70184816406532019097135596160, 32.79324138370125404493943352195, 33.09869394216564081159271016679
