L(s) = 1 | + (−0.5 − 0.866i)2-s + (−0.5 + 0.866i)3-s + (−0.5 + 0.866i)4-s + (−0.5 − 0.866i)5-s + 6-s + 7-s + 8-s + (−0.5 − 0.866i)9-s + (−0.5 + 0.866i)10-s + (−0.5 − 0.866i)11-s + (−0.5 − 0.866i)12-s + (−0.5 − 0.866i)13-s + (−0.5 − 0.866i)14-s + 15-s + (−0.5 − 0.866i)16-s + (−0.5 + 0.866i)17-s + ⋯ |
L(s) = 1 | + (−0.5 − 0.866i)2-s + (−0.5 + 0.866i)3-s + (−0.5 + 0.866i)4-s + (−0.5 − 0.866i)5-s + 6-s + 7-s + 8-s + (−0.5 − 0.866i)9-s + (−0.5 + 0.866i)10-s + (−0.5 − 0.866i)11-s + (−0.5 − 0.866i)12-s + (−0.5 − 0.866i)13-s + (−0.5 − 0.866i)14-s + 15-s + (−0.5 − 0.866i)16-s + (−0.5 + 0.866i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.317 + 0.948i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.317 + 0.948i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.2686623617 + 0.1933631880i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.2686623617 + 0.1933631880i\) |
\(L(1)\) |
\(\approx\) |
\(0.5285702855 - 0.1114330141i\) |
\(L(1)\) |
\(\approx\) |
\(0.5285702855 - 0.1114330141i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 1033 | \( 1 \) |
good | 2 | \( 1 + (-0.5 - 0.866i)T \) |
| 3 | \( 1 + (-0.5 + 0.866i)T \) |
| 5 | \( 1 + (-0.5 - 0.866i)T \) |
| 7 | \( 1 + T \) |
| 11 | \( 1 + (-0.5 - 0.866i)T \) |
| 13 | \( 1 + (-0.5 - 0.866i)T \) |
| 17 | \( 1 + (-0.5 + 0.866i)T \) |
| 19 | \( 1 + (-0.5 + 0.866i)T \) |
| 23 | \( 1 + (-0.5 + 0.866i)T \) |
| 29 | \( 1 + (-0.5 - 0.866i)T \) |
| 31 | \( 1 + (-0.5 + 0.866i)T \) |
| 37 | \( 1 + T \) |
| 41 | \( 1 + (-0.5 - 0.866i)T \) |
| 43 | \( 1 + (-0.5 - 0.866i)T \) |
| 47 | \( 1 + (-0.5 - 0.866i)T \) |
| 53 | \( 1 + T \) |
| 59 | \( 1 + (-0.5 + 0.866i)T \) |
| 61 | \( 1 + T \) |
| 67 | \( 1 + (-0.5 + 0.866i)T \) |
| 71 | \( 1 + (-0.5 + 0.866i)T \) |
| 73 | \( 1 + T \) |
| 79 | \( 1 + (-0.5 + 0.866i)T \) |
| 83 | \( 1 + (-0.5 + 0.866i)T \) |
| 89 | \( 1 + T \) |
| 97 | \( 1 + (-0.5 + 0.866i)T \) |
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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
−21.76464121652114764096521110483, −20.25411138639721134414621479765, −19.66656332370507396223532222784, −18.60747580127662025228911474957, −18.263303190181773724823969115749, −17.72271834992561037658811814898, −16.83269589585582780513832808035, −16.06224124588078623833919562879, −14.93778094545117762848626137220, −14.60482800345349738939854055832, −13.70606935985970759470699888337, −12.784572502349084071690171847039, −11.5622861669034873079148805356, −11.16447757379467148642282379947, −10.24799062011299634249303512847, −9.145408259942071172466901569673, −8.068598917240712132867054484824, −7.48900480438076032310179710278, −6.899411180708288242928020642254, −6.19120133460625668004572346561, −4.8666126301177091158283522964, −4.52682235335344699331739162090, −2.48213454447783254965675421192, −1.78535130375949941106128041273, −0.21320370750699259475660648743,
0.9839846671733492973778540461, 2.17674639244875017653304038516, 3.6056001324572358753956144208, 4.08121040576451357508810138676, 5.1357999738318630282963069793, 5.68264022390894670055870728694, 7.51070998302870429010559172881, 8.39485790219745842868014900956, 8.68546746487326352339237659938, 9.92984468616168741754378921473, 10.550298110099311533777649798634, 11.32713895780646177836870852531, 11.91557720541011511517582944514, 12.72328265589616807493960436800, 13.607712256703704737283910543136, 14.83257503307332642663887935609, 15.5753693672704779581052544362, 16.515205974168932261224020880607, 17.09618267895686443040494338251, 17.70783899912390261752272325080, 18.59041611971985786337911949584, 19.67653154446319486678297775348, 20.217862211170787210040346226855, 21.01519585765667227976841012543, 21.46364768758195805664002234979