L(s) = 1 | − 2-s + 4-s + 5-s − 8-s − 10-s − 11-s − 13-s + 16-s + 17-s − 19-s + 20-s + 22-s − 23-s + 25-s + 26-s − 29-s − 31-s − 32-s − 34-s + 37-s + 38-s − 40-s + 41-s + 43-s − 44-s + 46-s + 47-s + ⋯ |
L(s) = 1 | − 2-s + 4-s + 5-s − 8-s − 10-s − 11-s − 13-s + 16-s + 17-s − 19-s + 20-s + 22-s − 23-s + 25-s + 26-s − 29-s − 31-s − 32-s − 34-s + 37-s + 38-s − 40-s + 41-s + 43-s − 44-s + 46-s + 47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 21 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 21 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.4972623804\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.4972623804\) |
\(L(1)\) |
\(\approx\) |
\(0.6838072478\) |
\(L(1)\) |
\(\approx\) |
\(0.6838072478\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 7 | \( 1 \) |
good | 2 | \( 1 - T \) |
| 5 | \( 1 + T \) |
| 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
−39.50661155351630389171633460115, −38.21985593915273892734977795631, −36.87046584837538024852812190795, −36.286337297733217516955063033708, −34.455032738950987302447003018, −33.67499323055941205278829636710, −32.06905597513047654192922197302, −29.98408843411714656870834834775, −29.09345457094967826515164622089, −27.82592678096409045102785618249, −26.28746603707134008234353551778, −25.35180638669369017042649064483, −23.95175267810717684091926630693, −21.71236117436269543788775408707, −20.53983785320221716505090986872, −18.88189182634069320161874853493, −17.66389892894890481445748333359, −16.465154717935669189145654649009, −14.67082197914472783430852895830, −12.628728375934724403243638725586, −10.61054485467460396045181252678, −9.4646732073101131171217674859, −7.65463248516368468456673400912, −5.78036827357004952616082437239, −2.31518706430314115204629295971,
2.31518706430314115204629295971, 5.78036827357004952616082437239, 7.65463248516368468456673400912, 9.4646732073101131171217674859, 10.61054485467460396045181252678, 12.628728375934724403243638725586, 14.67082197914472783430852895830, 16.465154717935669189145654649009, 17.66389892894890481445748333359, 18.88189182634069320161874853493, 20.53983785320221716505090986872, 21.71236117436269543788775408707, 23.95175267810717684091926630693, 25.35180638669369017042649064483, 26.28746603707134008234353551778, 27.82592678096409045102785618249, 29.09345457094967826515164622089, 29.98408843411714656870834834775, 32.06905597513047654192922197302, 33.67499323055941205278829636710, 34.455032738950987302447003018, 36.286337297733217516955063033708, 36.87046584837538024852812190795, 38.21985593915273892734977795631, 39.50661155351630389171633460115