L(s) = 1 | + (−0.866 − 0.5i)2-s + (0.5 + 0.866i)4-s + (−0.866 + 0.5i)5-s + (−0.5 − 0.866i)7-s − i·8-s + 10-s + 11-s + (0.866 − 0.5i)13-s + i·14-s + (−0.5 + 0.866i)16-s + (0.866 + 0.5i)17-s + (0.866 − 0.5i)19-s + (−0.866 − 0.5i)20-s + (−0.866 − 0.5i)22-s − i·23-s + ⋯ |
L(s) = 1 | + (−0.866 − 0.5i)2-s + (0.5 + 0.866i)4-s + (−0.866 + 0.5i)5-s + (−0.5 − 0.866i)7-s − i·8-s + 10-s + 11-s + (0.866 − 0.5i)13-s + i·14-s + (−0.5 + 0.866i)16-s + (0.866 + 0.5i)17-s + (0.866 − 0.5i)19-s + (−0.866 − 0.5i)20-s + (−0.866 − 0.5i)22-s − i·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 111 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.660 - 0.751i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 111 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.660 - 0.751i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.5684304020 - 0.2572366731i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.5684304020 - 0.2572366731i\) |
\(L(1)\) |
\(\approx\) |
\(0.6500360196 - 0.1677609548i\) |
\(L(1)\) |
\(\approx\) |
\(0.6500360196 - 0.1677609548i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 37 | \( 1 \) |
good | 2 | \( 1 + (-0.866 - 0.5i)T \) |
| 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
−29.20959983482436955171515615211, −28.25151077375228398012760778458, −27.61571331104620474941945861825, −26.66203493015929699299821969428, −25.35594440628681708252368319151, −24.806197213455072316650058302101, −23.59451760834280649247302558936, −22.72091248115746086000581473008, −21.108238622779739204990581400540, −19.911080735722701558235340462187, −19.10788390422501729321144596004, −18.26176831346771909827634631623, −16.79974872506722154764564604789, −16.05286125730881233581925146780, −15.22694329059216373843796223895, −13.91664364351943800799541240392, −12.08404583900962286551502108469, −11.45812116959389169236555464627, −9.6937674437313361711511555048, −8.89297404906821733405471079743, −7.82034153624179659212658148004, −6.52363081314068441063054671311, −5.30509178758140489510305563571, −3.47476126712780434319198192278, −1.35716483168152416834975152809,
1.00359692796775783765040698108, 3.17379044843813013497596431807, 3.96754664809303985024975337678, 6.50067199164577397302951766379, 7.46042360023421027058728697200, 8.58310784701891324158397621601, 9.96785692350301540673965467322, 10.90995869649934195453978091381, 11.87435968556038700862077434397, 13.07908202064269521305487679569, 14.59343631079928711887973787603, 15.991939234159099305498758386084, 16.73961760973232605913782277528, 18.0064043850287485311365406371, 19.043730326038378647932339439925, 19.86098704176846053045942538044, 20.62254920656115659786200530266, 22.17930914562166505346572794079, 22.950176355126939543135235647584, 24.28551591755904625453199095480, 25.694176698601989912741380539670, 26.360316425861754075343660116899, 27.38883636336814032516047561555, 28.04119228279920814747060319134, 29.37018699383587379479411871434