L(s) = 1 | + (0.606 − 0.795i)2-s + (−0.973 + 0.227i)3-s + (−0.264 − 0.964i)4-s + (0.953 + 0.301i)5-s + (−0.409 + 0.912i)6-s + (0.477 − 0.878i)7-s + (−0.927 − 0.373i)8-s + (0.896 − 0.443i)9-s + (0.817 − 0.575i)10-s + (−0.997 + 0.0765i)11-s + (0.477 + 0.878i)12-s + (−0.114 − 0.993i)13-s + (−0.409 − 0.912i)14-s + (−0.997 − 0.0765i)15-s + (−0.859 + 0.511i)16-s + (0.338 − 0.941i)17-s + ⋯ |
L(s) = 1 | + (0.606 − 0.795i)2-s + (−0.973 + 0.227i)3-s + (−0.264 − 0.964i)4-s + (0.953 + 0.301i)5-s + (−0.409 + 0.912i)6-s + (0.477 − 0.878i)7-s + (−0.927 − 0.373i)8-s + (0.896 − 0.443i)9-s + (0.817 − 0.575i)10-s + (−0.997 + 0.0765i)11-s + (0.477 + 0.878i)12-s + (−0.114 − 0.993i)13-s + (−0.409 − 0.912i)14-s + (−0.997 − 0.0765i)15-s + (−0.859 + 0.511i)16-s + (0.338 − 0.941i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 83 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.0614 - 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 83 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.0614 - 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.7895317864 - 0.7424470195i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.7895317864 - 0.7424470195i\) |
\(L(1)\) |
\(\approx\) |
\(0.9975911307 - 0.5587075603i\) |
\(L(1)\) |
\(\approx\) |
\(0.9975911307 - 0.5587075603i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 83 | \( 1 \) |
good | 2 | \( 1 + (0.606 - 0.795i)T \) |
| 3 | \( 1 + (-0.973 + 0.227i)T \) |
| 5 | \( 1 + (0.953 + 0.301i)T \) |
| 7 | \( 1 + (0.477 - 0.878i)T \) |
| 11 | \( 1 + (-0.997 + 0.0765i)T \) |
| 13 | \( 1 + (-0.114 - 0.993i)T \) |
| 17 | \( 1 + (0.338 - 0.941i)T \) |
| 19 | \( 1 + (0.720 + 0.693i)T \) |
| 23 | \( 1 + (0.190 + 0.981i)T \) |
| 29 | \( 1 + (0.0383 + 0.999i)T \) |
| 31 | \( 1 + (-0.927 + 0.373i)T \) |
| 37 | \( 1 + (0.896 + 0.443i)T \) |
| 41 | \( 1 + (0.606 + 0.795i)T \) |
| 43 | \( 1 + (-0.771 - 0.636i)T \) |
| 47 | \( 1 + (-0.665 - 0.746i)T \) |
| 53 | \( 1 + (-0.665 + 0.746i)T \) |
| 59 | \( 1 + (0.988 + 0.152i)T \) |
| 61 | \( 1 + (-0.543 + 0.839i)T \) |
| 67 | \( 1 + (-0.859 + 0.511i)T \) |
| 71 | \( 1 + (0.477 + 0.878i)T \) |
| 73 | \( 1 + (0.817 - 0.575i)T \) |
| 79 | \( 1 + (-0.264 - 0.964i)T \) |
| 89 | \( 1 + (-0.409 + 0.912i)T \) |
| 97 | \( 1 + (-0.409 - 0.912i)T \) |
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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
−31.17328133797584968157802220727, −30.218251237693191163283225237721, −28.880170642231846366995121421165, −28.26773268740280062298081096118, −26.6698865092407965315231191865, −25.57333917358622361013257952252, −24.34507404811009144085575946877, −23.98476692854364021882195667726, −22.565322406625766258922143462066, −21.567563140947550479887127869938, −21.08839164223398411615772710109, −18.59689530139430400303656169381, −17.831298174808337759780401068279, −16.83687880334145824636311627050, −15.890153782450281360098695741233, −14.554419590122497685574797987190, −13.23820958151084314419460362281, −12.43151932988415095673816570485, −11.17096703130684239573885474189, −9.43372768223609974378785607145, −7.97951916178385709459947103137, −6.45345789679621682681852938987, −5.552763873224556349896920833574, −4.678330477860464171132922246865, −2.234323867643562009160844540027,
1.311350878960791353025002364376, 3.210851053244489387292216241540, 4.99115801727501699782245045092, 5.63515958486598203822476256850, 7.26811552572106525269699567040, 9.76542353764776577502103838723, 10.42466777647082708474971885151, 11.35505675975029442551953972872, 12.780895902144381518744266294629, 13.6834795394652609576263513851, 14.93338848867777083170902646402, 16.40108277795064491158493016353, 17.89626303120461330094831512461, 18.292091484876042174503375088988, 20.25484137897232134236280010858, 21.03571397662021660598692607091, 22.00464033271798999248724724750, 22.99227146183413127886806869791, 23.72658419651706267111894692741, 25.074047212236709390142246190223, 26.80969194611160712499113519319, 27.61720691064525064015142028684, 28.95155209956143248204349069667, 29.462291903563018525996423041, 30.251797905340018083881929299904