L(s) = 1 | − 2-s + 4-s − 5-s − 7-s − 8-s + 10-s + 11-s − 13-s + 14-s + 16-s + 17-s − 19-s − 20-s − 22-s + 23-s + 25-s + 26-s − 28-s − 29-s − 31-s − 32-s − 34-s + 35-s − 37-s + 38-s + 40-s − 41-s + ⋯ |
L(s) = 1 | − 2-s + 4-s − 5-s − 7-s − 8-s + 10-s + 11-s − 13-s + 14-s + 16-s + 17-s − 19-s − 20-s − 22-s + 23-s + 25-s + 26-s − 28-s − 29-s − 31-s − 32-s − 34-s + 35-s − 37-s + 38-s + 40-s − 41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 723 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 723 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.4618098119\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.4618098119\) |
\(L(1)\) |
\(\approx\) |
\(0.4673483517\) |
\(L(1)\) |
\(\approx\) |
\(0.4673483517\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 241 | \( 1 \) |
good | 2 | \( 1 - T \) |
| 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
−22.3801140479306106439665261291, −21.4729209256039889132065560932, −20.314083180893891270281766768874, −19.74833595013161681890656205449, −19.01875282375358722680616145320, −18.713549821969878327667615109572, −17.11893964991191721120575938389, −16.85595124767436654497346828797, −16.024338638692884454326263059108, −15.03667599251513102686365000720, −14.59394722015952270198362922667, −12.87554393992156189311940837420, −12.2440316283368091439283208376, −11.4878121952881518207622208904, −10.55272304473681005038290413907, −9.63333456601657292214318870884, −8.96127812852369975666219733270, −7.99245807143457710500199889688, −7.05368134361638534773436463989, −6.57519615159612360890896484333, −5.217282183980402878294928800127, −3.73310417936947975199329896831, −3.10439354757593737191662783361, −1.71478879680033717690255415475, −0.38686937177436045985636347702,
0.38686937177436045985636347702, 1.71478879680033717690255415475, 3.10439354757593737191662783361, 3.73310417936947975199329896831, 5.217282183980402878294928800127, 6.57519615159612360890896484333, 7.05368134361638534773436463989, 7.99245807143457710500199889688, 8.96127812852369975666219733270, 9.63333456601657292214318870884, 10.55272304473681005038290413907, 11.4878121952881518207622208904, 12.2440316283368091439283208376, 12.87554393992156189311940837420, 14.59394722015952270198362922667, 15.03667599251513102686365000720, 16.024338638692884454326263059108, 16.85595124767436654497346828797, 17.11893964991191721120575938389, 18.713549821969878327667615109572, 19.01875282375358722680616145320, 19.74833595013161681890656205449, 20.314083180893891270281766768874, 21.4729209256039889132065560932, 22.3801140479306106439665261291