L(s) = 1 | + (−0.281 − 0.959i)3-s + (0.989 − 0.142i)7-s + (−0.841 + 0.540i)9-s + (0.415 − 0.909i)11-s + (0.989 + 0.142i)13-s + (−0.755 + 0.654i)17-s + (0.654 − 0.755i)19-s + (−0.415 − 0.909i)21-s + (0.755 + 0.654i)27-s + (0.654 + 0.755i)29-s + (0.959 + 0.281i)31-s + (−0.989 − 0.142i)33-s + (−0.540 − 0.841i)37-s + (−0.142 − 0.989i)39-s + (0.841 + 0.540i)41-s + ⋯ |
L(s) = 1 | + (−0.281 − 0.959i)3-s + (0.989 − 0.142i)7-s + (−0.841 + 0.540i)9-s + (0.415 − 0.909i)11-s + (0.989 + 0.142i)13-s + (−0.755 + 0.654i)17-s + (0.654 − 0.755i)19-s + (−0.415 − 0.909i)21-s + (0.755 + 0.654i)27-s + (0.654 + 0.755i)29-s + (0.959 + 0.281i)31-s + (−0.989 − 0.142i)33-s + (−0.540 − 0.841i)37-s + (−0.142 − 0.989i)39-s + (0.841 + 0.540i)41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 460 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.201 - 0.979i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 460 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.201 - 0.979i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.722469035 - 1.404849850i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.722469035 - 1.404849850i\) |
\(L(1)\) |
\(\approx\) |
\(1.128924261 - 0.4420934950i\) |
\(L(1)\) |
\(\approx\) |
\(1.128924261 - 0.4420934950i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 23 | \( 1 \) |
good | 3 | \( 1 + (-0.281 - 0.959i)T \) |
| 7 | \( 1 + (0.989 - 0.142i)T \) |
| 11 | \( 1 + (0.415 - 0.909i)T \) |
| 13 | \( 1 + (0.989 + 0.142i)T \) |
| 17 | \( 1 + (-0.755 + 0.654i)T \) |
| 19 | \( 1 + (0.654 - 0.755i)T \) |
| 29 | \( 1 + (0.654 + 0.755i)T \) |
| 31 | \( 1 + (0.959 + 0.281i)T \) |
| 37 | \( 1 + (-0.540 - 0.841i)T \) |
| 41 | \( 1 + (0.841 + 0.540i)T \) |
| 43 | \( 1 + (0.281 + 0.959i)T \) |
| 47 | \( 1 - iT \) |
| 53 | \( 1 + (0.989 - 0.142i)T \) |
| 59 | \( 1 + (-0.142 + 0.989i)T \) |
| 61 | \( 1 + (0.959 + 0.281i)T \) |
| 67 | \( 1 + (-0.909 + 0.415i)T \) |
| 71 | \( 1 + (-0.415 - 0.909i)T \) |
| 73 | \( 1 + (0.755 + 0.654i)T \) |
| 79 | \( 1 + (0.142 - 0.989i)T \) |
| 83 | \( 1 + (-0.540 - 0.841i)T \) |
| 89 | \( 1 + (-0.959 + 0.281i)T \) |
| 97 | \( 1 + (0.540 - 0.841i)T \) |
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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
−23.793377776048999029889370733903, −22.7565120195822994915392681096, −22.39803715209634532011515644404, −21.06796018992191838792675690435, −20.76611976626871681642495690414, −19.945707043045773146819136037186, −18.56430567664602999015087984099, −17.68005992638796017896870132458, −17.15533082448091121385047926459, −15.87755293980802205728835920898, −15.43318835075299683968200748378, −14.41234473273486183063350817063, −13.69110440679981178009173033504, −12.15693104730795860138776866549, −11.54192088658952981204469508928, −10.66082000629493181630164529641, −9.74810710952539309739539400072, −8.83102424874639915353254337245, −7.92572748978956123525025414178, −6.59678224786110487387704920271, −5.522500045291112697164961793003, −4.61089103098283398072411896537, −3.85430024390309142232774702497, −2.444132819439885843779628055260, −1.0291102585581778146318942882,
0.77896324387333652368777952445, 1.58580745932808278576835709424, 2.889532553677572936903497795236, 4.263473755092622357158503381337, 5.46105699152772398335211158294, 6.3461360972811955055203542514, 7.25422074197532121771698491981, 8.39054368181311099409978549878, 8.82038904560518243467212830161, 10.63814823741695315661868958619, 11.269535847912966594622321403, 11.95704874025571705220803150165, 13.1968387673775091000268757093, 13.79067068424199932322038607425, 14.59409226011569979932259468536, 15.85887762628238330000915828154, 16.76735195359757342056656884466, 17.83161260260119094075346332682, 18.07869499009348127532293627206, 19.30658551020872776680630511976, 19.866097531867494513958376924722, 21.04660308625779528108959295001, 21.808482091725337948003847868582, 22.86071968472963784866394089429, 23.68874497575321845060939383953