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 + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 29 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 29 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.4657396651\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.4657396651\) |
\(L(1)\) |
\(\approx\) |
\(0.6117662895\) |
\(L(1)\) |
\(\approx\) |
\(0.6117662895\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 29 | \( 1 \) |
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 | \( 1 + T \) |
| 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
−37.14660426045612608592774008930, −36.13025701533360897988206562729, −34.759097299338730812239604825148, −33.713290308146147432370245351397, −33.12082197101496094180150219340, −30.58932128259149606033586979164, −29.354382967616109726243743295460, −28.51519021063355365846929501319, −27.4535382830190679798435094436, −26.08155139357491946652948772394, −24.698129503089230104650978934238, −23.547316535114245239312127860119, −21.53989598716732091875171728310, −20.75659856447482979112713688701, −18.50778885467398393360472019015, −17.81102604346635775963157582321, −16.78312280030606466990182922981, −15.292358015439331185195448880791, −13.07439942649190669490767093566, −11.19518919966302935918251101493, −10.41960537862811549543736767944, −8.6305478601229218895353492512, −6.75798544539269780881062624194, −5.316636251071941922447217297715, −1.79380926972434260937039337547,
1.79380926972434260937039337547, 5.316636251071941922447217297715, 6.75798544539269780881062624194, 8.6305478601229218895353492512, 10.41960537862811549543736767944, 11.19518919966302935918251101493, 13.07439942649190669490767093566, 15.292358015439331185195448880791, 16.78312280030606466990182922981, 17.81102604346635775963157582321, 18.50778885467398393360472019015, 20.75659856447482979112713688701, 21.53989598716732091875171728310, 23.547316535114245239312127860119, 24.698129503089230104650978934238, 26.08155139357491946652948772394, 27.4535382830190679798435094436, 28.51519021063355365846929501319, 29.354382967616109726243743295460, 30.58932128259149606033586979164, 33.12082197101496094180150219340, 33.713290308146147432370245351397, 34.759097299338730812239604825148, 36.13025701533360897988206562729, 37.14660426045612608592774008930