L(s) = 1 | + (0.866 + 0.5i)2-s + (0.5 + 0.866i)4-s + (−0.866 − 0.5i)7-s + i·8-s + (0.5 − 0.866i)11-s + (−0.866 + 0.5i)13-s + (−0.5 − 0.866i)14-s + (−0.5 + 0.866i)16-s − i·17-s − 19-s + (0.866 − 0.5i)22-s + (0.866 − 0.5i)23-s − 26-s − i·28-s + (−0.5 + 0.866i)29-s + ⋯ |
L(s) = 1 | + (0.866 + 0.5i)2-s + (0.5 + 0.866i)4-s + (−0.866 − 0.5i)7-s + i·8-s + (0.5 − 0.866i)11-s + (−0.866 + 0.5i)13-s + (−0.5 − 0.866i)14-s + (−0.5 + 0.866i)16-s − i·17-s − 19-s + (0.866 − 0.5i)22-s + (0.866 − 0.5i)23-s − 26-s − i·28-s + (−0.5 + 0.866i)29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 45 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.746 + 0.665i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 45 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.746 + 0.665i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.121634195 + 0.4273851470i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.121634195 + 0.4273851470i\) |
\(L(1)\) |
\(\approx\) |
\(1.325722943 + 0.3763795750i\) |
\(L(1)\) |
\(\approx\) |
\(1.325722943 + 0.3763795750i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 5 | \( 1 \) |
good | 2 | \( 1 + (0.866 + 0.5i)T \) |
| 7 | \( 1 + (-0.866 - 0.5i)T \) |
| 11 | \( 1 + (0.5 - 0.866i)T \) |
| 13 | \( 1 + (-0.866 + 0.5i)T \) |
| 17 | \( 1 - iT \) |
| 19 | \( 1 - T \) |
| 23 | \( 1 + (0.866 - 0.5i)T \) |
| 29 | \( 1 + (-0.5 + 0.866i)T \) |
| 31 | \( 1 + (-0.5 - 0.866i)T \) |
| 37 | \( 1 + iT \) |
| 41 | \( 1 + (0.5 + 0.866i)T \) |
| 43 | \( 1 + (0.866 + 0.5i)T \) |
| 47 | \( 1 + (0.866 + 0.5i)T \) |
| 53 | \( 1 + iT \) |
| 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 | \( 1 - iT \) |
| 79 | \( 1 + (0.5 - 0.866i)T \) |
| 83 | \( 1 + (-0.866 - 0.5i)T \) |
| 89 | \( 1 + T \) |
| 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
−34.00424710182707550064941062131, −32.706828236563411129643526471574, −31.94992297959395786946267085022, −30.765176794437272535741646955643, −29.682853874499062845660019833697, −28.61826334367469613993085645601, −27.59103933300739950094295491547, −25.68879787270294708197101095750, −24.72183463160220183854718004095, −23.26671817717762986731123883769, −22.35187144111126316973226425094, −21.3221358381869911538380189064, −19.83993267788212567375159670710, −19.0942840596746855877614191987, −17.22470879883436590376710263106, −15.51854160130533112352736896324, −14.65619229002377086451959002444, −12.933750872201971845870702291323, −12.2586821272709000813693138281, −10.57842653171016829805642816678, −9.33138809756658955992612103916, −6.981415230962860811795010775574, −5.58872653978057000220107781450, −3.928016253803500703636670860979, −2.27801123733464486006762319605,
2.95556757703445041003603393133, 4.46770315626047345629366907824, 6.19824974543228993050043173011, 7.32091131843316737551159368428, 9.14276701251768334339005258195, 11.07297458220718648110346385663, 12.50645745592483848194158297560, 13.65955531716316586034419374885, 14.79628171318518435844216429572, 16.32354548153577964310638340464, 17.00476420403583707246790406379, 19.02740114767636611270912008177, 20.34043452267309477114702265649, 21.74422327324655778294265741098, 22.65417322400857212273126914082, 23.8332839457911888648774790342, 24.922872477428019857450742488324, 26.10108139965325163671333111158, 27.16101443131673720727628981936, 29.241168675169633748227985056820, 29.75946837397514624139563632631, 31.34730036269974033254106837475, 32.17900424643717099887851175528, 33.164274991087520294780567206496, 34.35120427433898782995849327214