L(s) = 1 | + (−0.707 + 0.707i)3-s + (−0.707 − 0.707i)5-s − i·9-s + (0.707 + 0.707i)11-s + (−0.707 + 0.707i)13-s + 15-s + 17-s + (0.707 − 0.707i)19-s + i·23-s + i·25-s + (0.707 + 0.707i)27-s + (−0.707 + 0.707i)29-s + 31-s − 33-s + (0.707 + 0.707i)37-s + ⋯ |
L(s) = 1 | + (−0.707 + 0.707i)3-s + (−0.707 − 0.707i)5-s − i·9-s + (0.707 + 0.707i)11-s + (−0.707 + 0.707i)13-s + 15-s + 17-s + (0.707 − 0.707i)19-s + i·23-s + i·25-s + (0.707 + 0.707i)27-s + (−0.707 + 0.707i)29-s + 31-s − 33-s + (0.707 + 0.707i)37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 224 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.555 + 0.831i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 224 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.555 + 0.831i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.7041456197 + 0.3763736750i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.7041456197 + 0.3763736750i\) |
\(L(1)\) |
\(\approx\) |
\(0.7714055603 + 0.1752713426i\) |
\(L(1)\) |
\(\approx\) |
\(0.7714055603 + 0.1752713426i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 7 | \( 1 \) |
good | 3 | \( 1 + (-0.707 + 0.707i)T \) |
| 5 | \( 1 + (-0.707 - 0.707i)T \) |
| 11 | \( 1 + (0.707 + 0.707i)T \) |
| 13 | \( 1 + (-0.707 + 0.707i)T \) |
| 17 | \( 1 + T \) |
| 19 | \( 1 + (0.707 - 0.707i)T \) |
| 23 | \( 1 + iT \) |
| 29 | \( 1 + (-0.707 + 0.707i)T \) |
| 31 | \( 1 + T \) |
| 37 | \( 1 + (0.707 + 0.707i)T \) |
| 41 | \( 1 - iT \) |
| 43 | \( 1 + (0.707 + 0.707i)T \) |
| 47 | \( 1 - T \) |
| 53 | \( 1 + (-0.707 - 0.707i)T \) |
| 59 | \( 1 + (0.707 + 0.707i)T \) |
| 61 | \( 1 + (0.707 - 0.707i)T \) |
| 67 | \( 1 + (0.707 - 0.707i)T \) |
| 71 | \( 1 - iT \) |
| 73 | \( 1 - iT \) |
| 79 | \( 1 + T \) |
| 83 | \( 1 + (0.707 - 0.707i)T \) |
| 89 | \( 1 - iT \) |
| 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
−26.50219192833942226409911145889, −25.08129808316934753868074335315, −24.46467264713009920602524395490, −23.43055173901658357710228260949, −22.56924441160975745626127409787, −22.13374236347352256584190130332, −20.6150890266785909456986798433, −19.31432200221626655635355342497, −18.92746620204390747422840423261, −17.8884540179846142153017257940, −16.87413435734086149860473066504, −16.03311379769907920229508251144, −14.71486964392655910573279799789, −13.926050593516021841048649078714, −12.51489246678784335251515615646, −11.86449758606426675890430052674, −10.94288360585096408252238287322, −9.94709898089309447863058336306, −8.1810537265539234573700034268, −7.465792559537697753929223227086, −6.386634657177452339126750745, −5.41060242718449337447479859238, −3.86736150936024432793533329182, −2.57603458010095088720656535270, −0.78537340950087120718937157029,
1.22760792963433838039895572634, 3.39345166518824661224970858399, 4.506217389279392412442867126446, 5.20810539255516691076234871613, 6.6659496829359099577923325169, 7.78397827994218375366280185135, 9.30055695772888926750721959921, 9.77492005946730277289528905007, 11.416391492180130359241381282598, 11.84153729245017915650825285670, 12.84943829578295235630834618375, 14.44026538242305198771851465444, 15.304856209399394120011497354548, 16.2933661297669095334041998973, 16.94252899145532130455660876383, 17.851022222345333118095634912574, 19.27014424221249755313232098758, 20.10630363497358275544415953452, 21.05174048390574372163184524387, 21.98260957495937713833919971567, 22.90284531407045992258106529447, 23.70628537413281900039785336586, 24.534083245270655381119201385910, 25.79917401321993203636115334236, 26.85941261109527943086457391902