L(s) = 1 | + (0.995 − 0.0950i)2-s + (0.981 − 0.189i)4-s + (−0.723 + 0.690i)5-s + (−0.786 + 0.618i)7-s + (0.959 − 0.281i)8-s + (−0.654 + 0.755i)10-s + (0.981 − 0.189i)13-s + (−0.723 + 0.690i)14-s + (0.928 − 0.371i)16-s + (−0.841 − 0.540i)17-s + (0.841 − 0.540i)19-s + (−0.580 + 0.814i)20-s + (0.786 + 0.618i)23-s + (0.0475 − 0.998i)25-s + (0.959 − 0.281i)26-s + ⋯ |
L(s) = 1 | + (0.995 − 0.0950i)2-s + (0.981 − 0.189i)4-s + (−0.723 + 0.690i)5-s + (−0.786 + 0.618i)7-s + (0.959 − 0.281i)8-s + (−0.654 + 0.755i)10-s + (0.981 − 0.189i)13-s + (−0.723 + 0.690i)14-s + (0.928 − 0.371i)16-s + (−0.841 − 0.540i)17-s + (0.841 − 0.540i)19-s + (−0.580 + 0.814i)20-s + (0.786 + 0.618i)23-s + (0.0475 − 0.998i)25-s + (0.959 − 0.281i)26-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1089 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.629 + 0.777i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1089 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.629 + 0.777i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(3.271441008 + 1.560107710i\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.271441008 + 1.560107710i\) |
\(L(1)\) |
\(\approx\) |
\(1.769128479 + 0.2471459752i\) |
\(L(1)\) |
\(\approx\) |
\(1.769128479 + 0.2471459752i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 11 | \( 1 \) |
good | 2 | \( 1 + (0.995 - 0.0950i)T \) |
| 5 | \( 1 + (-0.723 + 0.690i)T \) |
| 7 | \( 1 + (-0.786 + 0.618i)T \) |
| 13 | \( 1 + (0.981 - 0.189i)T \) |
| 17 | \( 1 + (-0.841 - 0.540i)T \) |
| 19 | \( 1 + (0.841 - 0.540i)T \) |
| 23 | \( 1 + (0.786 + 0.618i)T \) |
| 29 | \( 1 + (0.888 + 0.458i)T \) |
| 31 | \( 1 + (-0.327 - 0.945i)T \) |
| 37 | \( 1 + (-0.654 + 0.755i)T \) |
| 41 | \( 1 + (-0.580 - 0.814i)T \) |
| 43 | \( 1 + (0.723 + 0.690i)T \) |
| 47 | \( 1 + (-0.580 + 0.814i)T \) |
| 53 | \( 1 + (0.142 + 0.989i)T \) |
| 59 | \( 1 + (0.995 + 0.0950i)T \) |
| 61 | \( 1 + (0.580 - 0.814i)T \) |
| 67 | \( 1 + (0.580 + 0.814i)T \) |
| 71 | \( 1 + (-0.841 + 0.540i)T \) |
| 73 | \( 1 + (-0.142 + 0.989i)T \) |
| 79 | \( 1 + (0.235 + 0.971i)T \) |
| 83 | \( 1 + (0.786 - 0.618i)T \) |
| 89 | \( 1 + (-0.841 - 0.540i)T \) |
| 97 | \( 1 + (0.723 + 0.690i)T \) |
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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
−20.94304572508477435553922343640, −20.58892713201389214657111351890, −19.62040057036793659763863420735, −19.30217663845208586650927957441, −17.93670269755190495317249350124, −16.84214899864553217715649884045, −16.21877769576419325316449491761, −15.7905049386895258376160597459, −14.90585508969185741029171177390, −13.86932353442129545966023384446, −13.244159388716663561375981936653, −12.6014823523865531861379304299, −11.82358660212780861375274506213, −10.97393294058011760436857517280, −10.227857145796605869257261380999, −8.91741420203454486507310406797, −8.1611754987375419091536833230, −7.10060958404482793110490866375, −6.51282426668829885619735022169, −5.46836196836069175226387395702, −4.53130132809181947839579565832, −3.771273536309019525003012491324, −3.17134416476620670733281856909, −1.70444143258370405051093170858, −0.61993594321354877085234165263,
0.93306903846178468070750507194, 2.4575279389270911687162587456, 3.11453838127411046406712989073, 3.8138191406634121757627561720, 4.89306243594224476536763353739, 5.838714431470338681054251134717, 6.69742067626145576094072084915, 7.23325686291279330483678116328, 8.39191867388689110721962129106, 9.44301783806023546190825139076, 10.495191338739115384143996929165, 11.33606321966611135102921298009, 11.75411482227340005564010983444, 12.79991332469734742378771550240, 13.43364010380192494285146451336, 14.2480184882799761116630866050, 15.23045716365359840363303133541, 15.78161400829306027779793383991, 16.08998921844216438268884734844, 17.48526566703859882107184802948, 18.533862211438293493781281694612, 19.115535396570958939053182461143, 19.924374706373687041016586171977, 20.56322873704734667622146056391, 21.584823636311649146413563561991