L(s) = 1 | + (−0.866 − 0.5i)2-s + (0.5 + 0.866i)3-s + (0.5 + 0.866i)4-s − i·6-s − i·7-s − i·8-s + (−0.5 + 0.866i)9-s + i·11-s + (−0.5 + 0.866i)12-s + (−0.5 + 0.866i)14-s + (−0.5 + 0.866i)16-s + 17-s + (0.866 − 0.5i)18-s − i·19-s + (0.866 − 0.5i)21-s + (0.5 − 0.866i)22-s + ⋯ |
L(s) = 1 | + (−0.866 − 0.5i)2-s + (0.5 + 0.866i)3-s + (0.5 + 0.866i)4-s − i·6-s − i·7-s − i·8-s + (−0.5 + 0.866i)9-s + i·11-s + (−0.5 + 0.866i)12-s + (−0.5 + 0.866i)14-s + (−0.5 + 0.866i)16-s + 17-s + (0.866 − 0.5i)18-s − i·19-s + (0.866 − 0.5i)21-s + (0.5 − 0.866i)22-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2015 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.930 - 0.366i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2015 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.930 - 0.366i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.02216754217 + 0.1167314794i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.02216754217 + 0.1167314794i\) |
\(L(1)\) |
\(\approx\) |
\(0.7433562288 + 0.08737959191i\) |
\(L(1)\) |
\(\approx\) |
\(0.7433562288 + 0.08737959191i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 \) |
| 13 | \( 1 \) |
| 31 | \( 1 \) |
good | 2 | \( 1 + (-0.866 - 0.5i)T \) |
| 3 | \( 1 + (0.5 + 0.866i)T \) |
| 7 | \( 1 - iT \) |
| 11 | \( 1 + iT \) |
| 17 | \( 1 + T \) |
| 19 | \( 1 - iT \) |
| 23 | \( 1 + (-0.5 + 0.866i)T \) |
| 29 | \( 1 + (-0.5 + 0.866i)T \) |
| 37 | \( 1 + (-0.866 + 0.5i)T \) |
| 41 | \( 1 - iT \) |
| 43 | \( 1 + T \) |
| 47 | \( 1 - iT \) |
| 53 | \( 1 + (0.5 + 0.866i)T \) |
| 59 | \( 1 + iT \) |
| 61 | \( 1 + (-0.5 - 0.866i)T \) |
| 67 | \( 1 + iT \) |
| 71 | \( 1 + (0.866 + 0.5i)T \) |
| 73 | \( 1 + (-0.866 + 0.5i)T \) |
| 79 | \( 1 + (-0.5 + 0.866i)T \) |
| 83 | \( 1 + (0.866 - 0.5i)T \) |
| 89 | \( 1 + (-0.866 - 0.5i)T \) |
| 97 | \( 1 + (0.866 - 0.5i)T \) |
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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
−19.10737921785734248009044960586, −18.63146133109239539380948790581, −18.15158498482151182153265094149, −17.26535534998758445055475007195, −16.45558500367901986133287490568, −15.85000667673538380425961240441, −14.84068660647517696703161277917, −14.421139836204144934313041232567, −13.69702011921581846965340686685, −12.58495670758995904485379059403, −11.97816152459877182374308229425, −11.23487388448497828634772744338, −10.21539023914919512280666372885, −9.35865626998091817635001724647, −8.72512749049498403244077584535, −8.00659385103098932876498526266, −7.6282622655722536836719144187, −6.272081149902801937383821383251, −6.099567842360600806444902385604, −5.215644699181601336240813701708, −3.61818762120042015074349889771, −2.685691739064724875613067227033, −1.92014559892431196373809108204, −1.009092697873219871540539268239, −0.02796915187059384922993864944,
1.218264347269876749677224864125, 2.137964211184647818781487178801, 3.141095378572569979711594988459, 3.82141797337340200729652868693, 4.55121765340423245448249652218, 5.58701975719457655802734874323, 7.16559571427561317418286979268, 7.32888471750774143325895553720, 8.35208525375581918661204619776, 9.13870729633054543575488410287, 9.84602869813980491400258960291, 10.32223035664757052172939797246, 10.99754854744297407286757902631, 11.83370798250572798935695074559, 12.72809977180384943759963005425, 13.59537568611220453034646422789, 14.30869841199811395574232309295, 15.30636810804475845936789117067, 15.82741731961071040679348992372, 16.72567358595278516462008597092, 17.195308327703295364585741481646, 17.884539588419623722105758300569, 18.883430136597694532733304967186, 19.66149860702406772494427421455, 20.13559411058335759886012852820