| 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 + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 73 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 73 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(1.741907430\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.741907430\) |
| \(L(1)\) |
\(\approx\) |
\(1.794636483\) |
| \(L(1)\) |
\(\approx\) |
\(1.794636483\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 73 | \( 1 \) |
| 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 | \( 1 + T \) |
| 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
−31.410952780342472027363132148771, −31.02736170697278072454381676429, −29.631583079970470978967117267366, −28.659981232933365653871089618185, −26.84786900747086139196535056632, −26.09180709583076451803961596351, −24.81571716970897524236893595400, −23.95287701989959187369445244593, −22.790404010198814653325005383796, −21.76766221918046631972443659677, −20.3432287181759549011681855680, −19.772275960075922407417433471963, −18.722161242359284092833550208002, −16.36169750261631524140625675010, −15.53955622545684648307412950081, −14.71339103427956601756490939006, −13.23662076809175706007294086306, −12.61039723044271894373865255190, −11.057600111831301161790763704769, −9.538462577126519392220210694474, −7.78599828751643495749040391811, −6.94757901318678836889055748081, −4.918936476482390378102427375243, −3.558510452056881428407611436422, −2.59808063262299142208016507350,
2.59808063262299142208016507350, 3.558510452056881428407611436422, 4.918936476482390378102427375243, 6.94757901318678836889055748081, 7.78599828751643495749040391811, 9.538462577126519392220210694474, 11.057600111831301161790763704769, 12.61039723044271894373865255190, 13.23662076809175706007294086306, 14.71339103427956601756490939006, 15.53955622545684648307412950081, 16.36169750261631524140625675010, 18.722161242359284092833550208002, 19.772275960075922407417433471963, 20.3432287181759549011681855680, 21.76766221918046631972443659677, 22.790404010198814653325005383796, 23.95287701989959187369445244593, 24.81571716970897524236893595400, 26.09180709583076451803961596351, 26.84786900747086139196535056632, 28.659981232933365653871089618185, 29.631583079970470978967117267366, 31.02736170697278072454381676429, 31.410952780342472027363132148771
