L(s) = 1 | + (0.866 − 0.5i)2-s + (−0.866 − 0.5i)3-s + (0.5 − 0.866i)4-s − 6-s − i·8-s + (0.5 + 0.866i)9-s + (−0.5 + 0.866i)11-s + (−0.866 + 0.5i)12-s + i·13-s + (−0.5 − 0.866i)16-s + (0.866 + 0.5i)17-s + (0.866 + 0.5i)18-s + (−0.5 − 0.866i)19-s + i·22-s + (−0.866 + 0.5i)23-s + (−0.5 + 0.866i)24-s + ⋯ |
L(s) = 1 | + (0.866 − 0.5i)2-s + (−0.866 − 0.5i)3-s + (0.5 − 0.866i)4-s − 6-s − i·8-s + (0.5 + 0.866i)9-s + (−0.5 + 0.866i)11-s + (−0.866 + 0.5i)12-s + i·13-s + (−0.5 − 0.866i)16-s + (0.866 + 0.5i)17-s + (0.866 + 0.5i)18-s + (−0.5 − 0.866i)19-s + i·22-s + (−0.866 + 0.5i)23-s + (−0.5 + 0.866i)24-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 35 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.413 - 0.910i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 35 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.413 - 0.910i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.7991983432 - 0.5145033684i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.7991983432 - 0.5145033684i\) |
\(L(1)\) |
\(\approx\) |
\(1.058335931 - 0.4733744821i\) |
\(L(1)\) |
\(\approx\) |
\(1.058335931 - 0.4733744821i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 \) |
| 7 | \( 1 \) |
good | 2 | \( 1 + (0.866 - 0.5i)T \) |
| 3 | \( 1 + (-0.866 - 0.5i)T \) |
| 11 | \( 1 + (-0.5 + 0.866i)T \) |
| 13 | \( 1 + iT \) |
| 17 | \( 1 + (0.866 + 0.5i)T \) |
| 19 | \( 1 + (-0.5 - 0.866i)T \) |
| 23 | \( 1 + (-0.866 + 0.5i)T \) |
| 29 | \( 1 - T \) |
| 31 | \( 1 + (0.5 - 0.866i)T \) |
| 37 | \( 1 + (0.866 - 0.5i)T \) |
| 41 | \( 1 - T \) |
| 43 | \( 1 - iT \) |
| 47 | \( 1 + (-0.866 + 0.5i)T \) |
| 53 | \( 1 + (0.866 + 0.5i)T \) |
| 59 | \( 1 + (-0.5 + 0.866i)T \) |
| 61 | \( 1 + (0.5 + 0.866i)T \) |
| 67 | \( 1 + (-0.866 - 0.5i)T \) |
| 71 | \( 1 + T \) |
| 73 | \( 1 + (-0.866 - 0.5i)T \) |
| 79 | \( 1 + (0.5 + 0.866i)T \) |
| 83 | \( 1 + iT \) |
| 89 | \( 1 + (-0.5 - 0.866i)T \) |
| 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
−35.749660366071901960504264976776, −34.4802257503679678150429883356, −33.93429512782188324977923086595, −32.524606221947250608406599823684, −31.84906682296229395054317923286, −30.08133720772457709176164179164, −29.21732571396782478724090916253, −27.60985463714245214892458399196, −26.40286115223966556390009930542, −24.912832447440480017061203137664, −23.62416837487569539975816556832, −22.71506978607732521960258885206, −21.59312202463794910859267667105, −20.55777022636353710996272154172, −18.271195512051699914532146571234, −16.81027988209426194463775545133, −15.94113780529554755218316419443, −14.62692708812760852913901544285, −12.97760753792570457240425600923, −11.72876029289555446369806732339, −10.33870006154565848158924862132, −8.03553049727166256414221079787, −6.19593039359663724126985999563, −5.138662754376125234002452406466, −3.43481525921369310600501690633,
1.969169021137100745218735244449, 4.392224696474392375853104380304, 5.83395042694476754865003442461, 7.24740272572690371684591513095, 9.97974127688517494176518911515, 11.35423713312006801274194634192, 12.42726294034201006771538127779, 13.573690714928856642771938895439, 15.20299177757423858075662843866, 16.71797000479940638888174294937, 18.33506531340939685212571873723, 19.52193822755942282423909543918, 21.11571848579420403991928649107, 22.22675845269073007084478239975, 23.4376316269784483465833604334, 24.10864515671436679354702270075, 25.7071536840550789463868600973, 27.91872212879256090594703309199, 28.60477526369126469121963508977, 29.84923286999480506900671721143, 30.71671367385168065015716871519, 32.0350327755604333546765305005, 33.51379303900399491664496161263, 34.158330518742277288403798722381, 35.80013473603395144869932470468