L(s) = 1 | + (0.477 + 0.878i)2-s + (−0.264 + 0.964i)3-s + (−0.543 + 0.839i)4-s + (−0.771 − 0.636i)5-s + (−0.973 + 0.227i)6-s + (−0.665 + 0.746i)7-s + (−0.997 − 0.0765i)8-s + (−0.859 − 0.511i)9-s + (0.190 − 0.981i)10-s + (0.817 + 0.575i)11-s + (−0.665 − 0.746i)12-s + (0.606 + 0.795i)13-s + (−0.973 − 0.227i)14-s + (0.817 − 0.575i)15-s + (−0.409 − 0.912i)16-s + (−0.927 − 0.373i)17-s + ⋯ |
L(s) = 1 | + (0.477 + 0.878i)2-s + (−0.264 + 0.964i)3-s + (−0.543 + 0.839i)4-s + (−0.771 − 0.636i)5-s + (−0.973 + 0.227i)6-s + (−0.665 + 0.746i)7-s + (−0.997 − 0.0765i)8-s + (−0.859 − 0.511i)9-s + (0.190 − 0.981i)10-s + (0.817 + 0.575i)11-s + (−0.665 − 0.746i)12-s + (0.606 + 0.795i)13-s + (−0.973 − 0.227i)14-s + (0.817 − 0.575i)15-s + (−0.409 − 0.912i)16-s + (−0.927 − 0.373i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 83 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.991 + 0.131i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 83 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.991 + 0.131i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.05218065242 + 0.7905776382i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.05218065242 + 0.7905776382i\) |
\(L(1)\) |
\(\approx\) |
\(0.5496656295 + 0.7235095732i\) |
\(L(1)\) |
\(\approx\) |
\(0.5496656295 + 0.7235095732i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 83 | \( 1 \) |
good | 2 | \( 1 + (0.477 + 0.878i)T \) |
| 3 | \( 1 + (-0.264 + 0.964i)T \) |
| 5 | \( 1 + (-0.771 - 0.636i)T \) |
| 7 | \( 1 + (-0.665 + 0.746i)T \) |
| 11 | \( 1 + (0.817 + 0.575i)T \) |
| 13 | \( 1 + (0.606 + 0.795i)T \) |
| 17 | \( 1 + (-0.927 - 0.373i)T \) |
| 19 | \( 1 + (0.988 + 0.152i)T \) |
| 23 | \( 1 + (0.0383 + 0.999i)T \) |
| 29 | \( 1 + (0.953 + 0.301i)T \) |
| 31 | \( 1 + (-0.997 + 0.0765i)T \) |
| 37 | \( 1 + (-0.859 + 0.511i)T \) |
| 41 | \( 1 + (0.477 - 0.878i)T \) |
| 43 | \( 1 + (0.720 + 0.693i)T \) |
| 47 | \( 1 + (0.896 - 0.443i)T \) |
| 53 | \( 1 + (0.896 + 0.443i)T \) |
| 59 | \( 1 + (0.338 - 0.941i)T \) |
| 61 | \( 1 + (-0.114 + 0.993i)T \) |
| 67 | \( 1 + (-0.409 - 0.912i)T \) |
| 71 | \( 1 + (-0.665 - 0.746i)T \) |
| 73 | \( 1 + (0.190 - 0.981i)T \) |
| 79 | \( 1 + (-0.543 + 0.839i)T \) |
| 89 | \( 1 + (-0.973 + 0.227i)T \) |
| 97 | \( 1 + (-0.973 - 0.227i)T \) |
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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
−30.40514451611457770568614286608, −29.53729545314569268195402922331, −28.61847969144331011893273800864, −27.38264193177434622904122654138, −26.317403499563308402102476685717, −24.66257547117342909971811242043, −23.65105681274879752057046251699, −22.725176695075658719030643369552, −22.22596310178353875622516273957, −20.20406999105459973221980505443, −19.62099945567106554926675163251, −18.70000994333034342166705577667, −17.63687227050735378908484626358, −15.974058909552629249071988880803, −14.4057382596461191680007147496, −13.48368430785477614493983758725, −12.434130208426622933102168016317, −11.309599950215721890344321137, −10.52340557319031042422876407128, −8.64711427042757425744173241379, −7.03517808962717517281606840274, −5.98576381108846083005965430241, −3.98833868750031452099225337312, −2.84246514862309621577873834592, −0.815236042323645389927492298540,
3.457954087571255053816876163914, 4.453525170522151707296270738730, 5.64267948340131258768671896026, 7.00391253568421978409093548518, 8.817547161371300351088891684422, 9.33042083177724841010798311817, 11.55782268038262499611970983485, 12.315893311403804686756918526690, 13.88459024725852851862490584727, 15.284468281628507499815297652017, 15.88109497274240867055448296638, 16.66806721130211144936694465684, 17.94524481925143987750908429803, 19.64420415891106456185444380196, 20.87154423013596017536839552712, 22.053013793509794789071508344479, 22.75149705034792571308536237128, 23.807131680817353843286211799890, 25.018432298325488997353779232, 26.00488102913491256066678660633, 27.12547700974422107958813404674, 27.94404088736116588195230033414, 29.007860524926009247717788310744, 31.171689524395863445924802047708, 31.31627910457141149261718208336