L(s) = 1 | + (0.766 − 0.642i)2-s + (−0.835 − 0.549i)3-s + (0.173 − 0.984i)4-s + (−0.396 + 0.918i)5-s + (−0.993 + 0.116i)6-s + (−0.993 + 0.116i)7-s + (−0.5 − 0.866i)8-s + (0.396 + 0.918i)9-s + (0.286 + 0.957i)10-s + (0.286 − 0.957i)11-s + (−0.686 + 0.727i)12-s + (0.597 + 0.802i)13-s + (−0.686 + 0.727i)14-s + (0.835 − 0.549i)15-s + (−0.939 − 0.342i)16-s + (0.766 − 0.642i)17-s + ⋯ |
L(s) = 1 | + (0.766 − 0.642i)2-s + (−0.835 − 0.549i)3-s + (0.173 − 0.984i)4-s + (−0.396 + 0.918i)5-s + (−0.993 + 0.116i)6-s + (−0.993 + 0.116i)7-s + (−0.5 − 0.866i)8-s + (0.396 + 0.918i)9-s + (0.286 + 0.957i)10-s + (0.286 − 0.957i)11-s + (−0.686 + 0.727i)12-s + (0.597 + 0.802i)13-s + (−0.686 + 0.727i)14-s + (0.835 − 0.549i)15-s + (−0.939 − 0.342i)16-s + (0.766 − 0.642i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.703 - 0.710i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.703 - 0.710i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.4832804062 - 1.158673025i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.4832804062 - 1.158673025i\) |
\(L(1)\) |
\(\approx\) |
\(0.8617576849 - 0.5148189136i\) |
\(L(1)\) |
\(\approx\) |
\(0.8617576849 - 0.5148189136i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 37 | \( 1 \) |
| 109 | \( 1 \) |
good | 2 | \( 1 + (0.766 - 0.642i)T \) |
| 3 | \( 1 + (-0.835 - 0.549i)T \) |
| 5 | \( 1 + (-0.396 + 0.918i)T \) |
| 7 | \( 1 + (-0.993 + 0.116i)T \) |
| 11 | \( 1 + (0.286 - 0.957i)T \) |
| 13 | \( 1 + (0.597 + 0.802i)T \) |
| 17 | \( 1 + (0.766 - 0.642i)T \) |
| 19 | \( 1 + (-0.5 - 0.866i)T \) |
| 23 | \( 1 + (-0.939 - 0.342i)T \) |
| 29 | \( 1 + (0.286 + 0.957i)T \) |
| 31 | \( 1 + (0.0581 + 0.998i)T \) |
| 41 | \( 1 + (0.939 + 0.342i)T \) |
| 43 | \( 1 + (0.939 - 0.342i)T \) |
| 47 | \( 1 + (-0.597 + 0.802i)T \) |
| 53 | \( 1 + (0.686 - 0.727i)T \) |
| 59 | \( 1 + (-0.835 + 0.549i)T \) |
| 61 | \( 1 + (0.286 + 0.957i)T \) |
| 67 | \( 1 + (-0.973 + 0.230i)T \) |
| 71 | \( 1 + (0.766 + 0.642i)T \) |
| 73 | \( 1 + (0.973 - 0.230i)T \) |
| 79 | \( 1 + (-0.286 + 0.957i)T \) |
| 83 | \( 1 + (0.396 - 0.918i)T \) |
| 89 | \( 1 + (0.0581 - 0.998i)T \) |
| 97 | \( 1 + (0.835 + 0.549i)T \) |
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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
−18.549663777948826005860531683524, −17.626635964079362866479952466090, −17.103125271863029282294944676459, −16.570657141381688742274811915724, −16.01693646405796564469843823556, −15.38356001124579797982993117131, −15.013238401423073292253637174061, −13.9217899186275802651960470255, −13.06043174332465646914684342836, −12.43007681093855788067861742052, −12.284941556424297023843276820613, −11.40387938637567169945422522748, −10.41364981986207932061263437977, −9.7423664508356483559433249471, −9.05670290637085286470913801001, −7.9807596476836060374295232684, −7.5945236831305108947355981096, −6.358747671385839411604069102467, −6.02975512591363104469372426840, −5.38457909453007874373913461278, −4.455628714644202462668858121520, −3.88442768311020817280462510644, −3.50307285327289340635869453602, −2.09359757702720964819384848323, −0.80479180177196529756889183281,
0.4147143116572207494787453294, 1.334843155077444740555387759421, 2.462705758133591996552982991298, 3.05275290995014132469836325781, 3.83707168729894827172069527386, 4.59175784222204799718413073494, 5.69040873996603741371668533, 6.16244373181534578240449530089, 6.75555150544111833538399896617, 7.26808140668829578518807963887, 8.53075891959295483426441117012, 9.42828719440897562396296754807, 10.3108949010266839908184482407, 10.8450856150403394809715102806, 11.41457013103095443432374356971, 12.02986755373006023411515447286, 12.58635304453105606228092643533, 13.42503479687244487332020619168, 14.00179724709623472826263330986, 14.48131751128630937671206400978, 15.66641418653908980219270566168, 16.11871632990768814561980463569, 16.56335843368134417688348659770, 17.91698608475611586465795415092, 18.34584994615829991553108044933