L(s) = 1 | + (−0.826 + 0.563i)3-s + (0.0747 + 0.997i)5-s + (0.365 − 0.930i)9-s + (−0.365 − 0.930i)11-s + (0.623 − 0.781i)13-s + (−0.623 − 0.781i)15-s + (−0.733 − 0.680i)17-s + (0.5 + 0.866i)19-s + (0.733 − 0.680i)23-s + (−0.988 + 0.149i)25-s + (0.222 + 0.974i)27-s + (−0.222 + 0.974i)29-s + (0.5 − 0.866i)31-s + (0.826 + 0.563i)33-s + (0.955 − 0.294i)37-s + ⋯ |
L(s) = 1 | + (−0.826 + 0.563i)3-s + (0.0747 + 0.997i)5-s + (0.365 − 0.930i)9-s + (−0.365 − 0.930i)11-s + (0.623 − 0.781i)13-s + (−0.623 − 0.781i)15-s + (−0.733 − 0.680i)17-s + (0.5 + 0.866i)19-s + (0.733 − 0.680i)23-s + (−0.988 + 0.149i)25-s + (0.222 + 0.974i)27-s + (−0.222 + 0.974i)29-s + (0.5 − 0.866i)31-s + (0.826 + 0.563i)33-s + (0.955 − 0.294i)37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 196 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.996 + 0.0853i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 196 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.996 + 0.0853i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.276442823 + 0.05459193518i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.276442823 + 0.05459193518i\) |
\(L(1)\) |
\(\approx\) |
\(0.8690025398 + 0.1385522120i\) |
\(L(1)\) |
\(\approx\) |
\(0.8690025398 + 0.1385522120i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 7 | \( 1 \) |
good | 3 | \( 1 + (-0.826 + 0.563i)T \) |
| 5 | \( 1 + (0.0747 + 0.997i)T \) |
| 11 | \( 1 + (-0.365 - 0.930i)T \) |
| 13 | \( 1 + (0.623 - 0.781i)T \) |
| 17 | \( 1 + (-0.733 - 0.680i)T \) |
| 19 | \( 1 + (0.5 + 0.866i)T \) |
| 23 | \( 1 + (0.733 - 0.680i)T \) |
| 29 | \( 1 + (-0.222 + 0.974i)T \) |
| 31 | \( 1 + (0.5 - 0.866i)T \) |
| 37 | \( 1 + (0.955 - 0.294i)T \) |
| 41 | \( 1 + (-0.900 - 0.433i)T \) |
| 43 | \( 1 + (0.900 - 0.433i)T \) |
| 47 | \( 1 + (0.988 + 0.149i)T \) |
| 53 | \( 1 + (0.955 + 0.294i)T \) |
| 59 | \( 1 + (-0.0747 + 0.997i)T \) |
| 61 | \( 1 + (0.955 - 0.294i)T \) |
| 67 | \( 1 + (0.5 - 0.866i)T \) |
| 71 | \( 1 + (0.222 + 0.974i)T \) |
| 73 | \( 1 + (-0.988 + 0.149i)T \) |
| 79 | \( 1 + (0.5 + 0.866i)T \) |
| 83 | \( 1 + (-0.623 - 0.781i)T \) |
| 89 | \( 1 + (0.365 - 0.930i)T \) |
| 97 | \( 1 + T \) |
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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
−26.82013228327676088735921719921, −25.57397321561474583131166160010, −24.70177402104853069886545047548, −23.76714826146620431427324575881, −23.261323931804525185556399446062, −22.005018218025896742242168935, −21.09018526861615843924448505033, −19.99848927937038870127785281782, −19.039935284646044397570502465700, −17.80645950330346609179635473768, −17.26862453922644688766375627998, −16.20590539591833531426242683144, −15.34534991519202414502712717151, −13.56276968384829498966526729854, −13.02414581188777592179594156451, −11.97383010386133952710254908418, −11.12933407554217858492676844827, −9.78624864460257684413840575535, −8.640487004332279422049276168469, −7.41961716330133724559004419453, −6.34700740504918092925804481528, −5.15173797457388261376056705211, −4.301144490947930005828247638, −2.10265775059842681071561978645, −0.955331453190650805810973605483,
0.6604713196202959750489026911, 2.81094346703827457547093678851, 3.88496183092765564692391349352, 5.413079092420108388112345203106, 6.20581243994193851073015799577, 7.36959500372648600073734390309, 8.84035182980653444223893880467, 10.17960170584714991489680163447, 10.86469089361055188553086962142, 11.626486234794137434820876098375, 13.01509660173145484154091978317, 14.16660013511240982904702606654, 15.29409215811757633414693951351, 16.03761747273386668819983681926, 17.09282164047353883968063184413, 18.27171186236908024391963587398, 18.64704537975250987530283022051, 20.29565043738167791076876203235, 21.20313412873088204409498056901, 22.24598619759568847244593677623, 22.725512447828793265867799905195, 23.69231164900046611091392937659, 24.859009534062802236306902686405, 26.06174045935363360237728695627, 26.93409699541458351930196020896