L(s) = 1 | + i·2-s − 3-s − 4-s − i·5-s − i·6-s + 7-s − i·8-s + 9-s + 10-s − 11-s + 12-s − i·13-s + i·14-s + i·15-s + 16-s − i·17-s + ⋯ |
L(s) = 1 | + i·2-s − 3-s − 4-s − i·5-s − i·6-s + 7-s − i·8-s + 9-s + 10-s − 11-s + 12-s − i·13-s + i·14-s + i·15-s + 16-s − i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 37 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.763 - 0.646i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 37 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.763 - 0.646i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.7412395992 - 0.2717602033i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.7412395992 - 0.2717602033i\) |
\(L(1)\) |
\(\approx\) |
\(0.7279167277 + 0.06024423023i\) |
\(L(1)\) |
\(\approx\) |
\(0.7279167277 + 0.06024423023i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 37 | \( 1 \) |
good | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 + iT \) |
| 7 | \( 1 - T \) |
| 11 | \( 1 - T \) |
| 13 | \( 1 - iT \) |
| 17 | \( 1 - iT \) |
| 19 | \( 1 + T \) |
| 23 | \( 1 - iT \) |
| 29 | \( 1 + T \) |
| 31 | \( 1 + T \) |
| 41 | \( 1 + T \) |
| 43 | \( 1 - iT \) |
| 47 | \( 1 + iT \) |
| 53 | \( 1 + iT \) |
| 59 | \( 1 + T \) |
| 61 | \( 1 - iT \) |
| 67 | \( 1 + iT \) |
| 71 | \( 1 - iT \) |
| 73 | \( 1 + iT \) |
| 79 | \( 1 - T \) |
| 83 | \( 1 - iT \) |
| 89 | \( 1 - iT \) |
| 97 | \( 1 + iT \) |
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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
−35.48889187906396478588624099561, −34.14116519199985661833693787526, −33.37121270690049889324078612615, −31.42351909459715401128376413813, −30.43695965808936739084556542574, −29.442690082954308381006983715816, −28.39385906756649842145484451930, −27.25813373171165244577455867558, −26.284933176202628419151752563869, −23.89416673663820769839409341499, −23.0757774706340658982311724958, −21.73293222992005180898211617588, −21.05488321184147374785129251114, −18.9988636491002231108017287498, −18.21529580774691769136889862836, −17.11826399161001533740919794276, −15.05212675722895903193399358668, −13.56770498249987919339825994882, −11.90892196243595279830171802447, −11.023208353879332346728659468945, −10.03631058252056740220159506585, −7.79307767942996972293028877074, −5.740424920407127445224915583427, −4.124712373221212554432271434645, −1.891032073729348382035324340261,
0.61549686593822271927427467532, 4.806537687650580890921692256715, 5.367288931553414114462919741812, 7.31228850687287547825191343163, 8.66050249025699236082624712455, 10.48657708887078487088265507824, 12.29163697425835362595040914215, 13.47247533957604169469695772660, 15.361026706308112161924320026520, 16.34459084233482959740358822823, 17.54819346805107344062027090053, 18.28297956348607433757262227092, 20.58675241728620076806689173337, 21.93490635749659673172741735351, 23.36159398125948959264241165815, 24.10757471078202921369630659360, 25.087815753345699111645577512740, 26.94437376469512416231180352300, 27.79159146636163699922054840962, 28.80444057999501108466474263887, 30.565618013227519280535083671370, 31.97737086532945481397552339744, 33.02334902128787778837942243335, 34.08580503751473052718054528432, 34.88877637373585496863803885232