L(s) = 1 | + (0.866 + 0.5i)2-s + (0.5 + 0.866i)3-s + (0.5 + 0.866i)4-s + (0.866 − 0.5i)5-s + i·6-s + (−0.5 − 0.866i)7-s + i·8-s + (−0.5 + 0.866i)9-s + 10-s − 11-s + (−0.5 + 0.866i)12-s + (0.866 − 0.5i)13-s − i·14-s + (0.866 + 0.5i)15-s + (−0.5 + 0.866i)16-s + (−0.866 − 0.5i)17-s + ⋯ |
L(s) = 1 | + (0.866 + 0.5i)2-s + (0.5 + 0.866i)3-s + (0.5 + 0.866i)4-s + (0.866 − 0.5i)5-s + i·6-s + (−0.5 − 0.866i)7-s + i·8-s + (−0.5 + 0.866i)9-s + 10-s − 11-s + (−0.5 + 0.866i)12-s + (0.866 − 0.5i)13-s − i·14-s + (0.866 + 0.5i)15-s + (−0.5 + 0.866i)16-s + (−0.866 − 0.5i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 37 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.320 + 0.947i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 37 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.320 + 0.947i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(2.214878642 + 1.588705544i\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.214878642 + 1.588705544i\) |
\(L(1)\) |
\(\approx\) |
\(1.788899329 + 0.8842390261i\) |
\(L(1)\) |
\(\approx\) |
\(1.788899329 + 0.8842390261i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 37 | \( 1 \) |
good | 2 | \( 1 + (0.866 + 0.5i)T \) |
| 3 | \( 1 + (0.5 + 0.866i)T \) |
| 5 | \( 1 + (0.866 - 0.5i)T \) |
| 7 | \( 1 + (-0.5 - 0.866i)T \) |
| 11 | \( 1 - T \) |
| 13 | \( 1 + (0.866 - 0.5i)T \) |
| 17 | \( 1 + (-0.866 - 0.5i)T \) |
| 19 | \( 1 + (0.866 - 0.5i)T \) |
| 23 | \( 1 + iT \) |
| 29 | \( 1 - iT \) |
| 31 | \( 1 - iT \) |
| 41 | \( 1 + (0.5 + 0.866i)T \) |
| 43 | \( 1 + iT \) |
| 47 | \( 1 + T \) |
| 53 | \( 1 + (-0.5 + 0.866i)T \) |
| 59 | \( 1 + (-0.866 - 0.5i)T \) |
| 61 | \( 1 + (-0.866 + 0.5i)T \) |
| 67 | \( 1 + (0.5 + 0.866i)T \) |
| 71 | \( 1 + (-0.5 - 0.866i)T \) |
| 73 | \( 1 - T \) |
| 79 | \( 1 + (0.866 - 0.5i)T \) |
| 83 | \( 1 + (-0.5 + 0.866i)T \) |
| 89 | \( 1 + (0.866 + 0.5i)T \) |
| 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
−35.01186838611360584385674063343, −33.70418417569581147698496796401, −32.44782608430526579586468002600, −31.169153920152052959432434091802, −30.56541542877876940899920486769, −28.981540388896056777444053422547, −28.77952340206667729225846412995, −26.15064172844240294351249134154, −25.185994030227197422839716216557, −24.07386467213252080654015074150, −22.74825255930772399367984767452, −21.53658318833374052599655366013, −20.39839012866768381106677004500, −18.84607377844038018197071116142, −18.20823105198513430425428023662, −15.735376192637147899707843847643, −14.34504630906104646115409071771, −13.342609028580464839831913035940, −12.35328010993638235746327216750, −10.658998884502228069219293882543, −8.99777412516335396985127923721, −6.7475769032492476719774631092, −5.6824733784314758163677025145, −3.10701424287336882713489175299, −1.937992876899654787260678737377,
2.879385196491709182319651624507, 4.484115760525524703752214971256, 5.82306433277190544103062121737, 7.79426069992086584105623131643, 9.45916157599402741819201088866, 10.95509100079841055132150098543, 13.33432963263049683323164711543, 13.63825153032010225657527703777, 15.47319045165383180818238903034, 16.30408133763966006073614798407, 17.61291363123304929658793610378, 20.17709483797286188836548309907, 20.838668062863841128854470877873, 22.0211830422638499961370973998, 23.206279951262925653279670161448, 24.71066482671482379292765836842, 25.85376882896272511826575381307, 26.5505104733957572481513935030, 28.4791402592070744765104733059, 29.72826054011695556284856289814, 31.16412993949204206322847061738, 32.21791630223314197995884199460, 33.109954340164283052452655188053, 33.65852692686942367308486846130, 35.5053932190607241685130514788