| 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 + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 839 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 839 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(8.830173694\) |
| \(L(\frac12)\) |
\(\approx\) |
\(8.830173694\) |
| \(L(1)\) |
\(\approx\) |
\(3.579174166\) |
| \(L(1)\) |
\(\approx\) |
\(3.579174166\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 839 | \( 1 \) |
| 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 | \( 1 - T \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−21.81655984368735001539255061277, −21.14065995776623039864209374378, −20.50588849301516776331442793630, −20.01935942054174052321748001000, −18.789784711096090649778393067303, −18.007279630893144433213681429130, −17.07197336332142024417404559984, −16.054708029800516933480138399694, −15.00029762637685695135844967730, −14.72039995733025617742107770638, −13.76288534992923670815284904673, −13.26198502570475074801626346297, −12.54593044528026151435677242251, −11.309800336000100378995928782539, −10.517926278511949078443576449094, −9.61664385078484306504865863881, −8.64013664794952330134158571025, −7.48045698722959667140575098868, −7.08208814128991682118894751367, −5.55245518670407891084565833525, −5.0395226911881876502251449075, −4.08338533492799588282148203643, −2.716288983938340633423786560291, −2.32646727608209715706610025730, −1.33434166339943289377494614627,
1.33434166339943289377494614627, 2.32646727608209715706610025730, 2.716288983938340633423786560291, 4.08338533492799588282148203643, 5.0395226911881876502251449075, 5.55245518670407891084565833525, 7.08208814128991682118894751367, 7.48045698722959667140575098868, 8.64013664794952330134158571025, 9.61664385078484306504865863881, 10.517926278511949078443576449094, 11.309800336000100378995928782539, 12.54593044528026151435677242251, 13.26198502570475074801626346297, 13.76288534992923670815284904673, 14.72039995733025617742107770638, 15.00029762637685695135844967730, 16.054708029800516933480138399694, 17.07197336332142024417404559984, 18.007279630893144433213681429130, 18.789784711096090649778393067303, 20.01935942054174052321748001000, 20.50588849301516776331442793630, 21.14065995776623039864209374378, 21.81655984368735001539255061277
