L(s) = 1 | + (0.906 + 0.421i)2-s + (0.399 + 0.916i)3-s + (0.644 + 0.764i)4-s + (0.215 − 0.976i)5-s + (−0.0241 + 0.999i)6-s + (0.568 − 0.822i)7-s + (0.262 + 0.964i)8-s + (−0.681 + 0.732i)9-s + (0.607 − 0.794i)10-s + (−0.443 + 0.896i)12-s + (0.568 − 0.822i)13-s + (0.861 − 0.506i)14-s + (0.981 − 0.192i)15-s + (−0.168 + 0.985i)16-s + (0.443 + 0.896i)17-s + (−0.926 + 0.377i)18-s + ⋯ |
L(s) = 1 | + (0.906 + 0.421i)2-s + (0.399 + 0.916i)3-s + (0.644 + 0.764i)4-s + (0.215 − 0.976i)5-s + (−0.0241 + 0.999i)6-s + (0.568 − 0.822i)7-s + (0.262 + 0.964i)8-s + (−0.681 + 0.732i)9-s + (0.607 − 0.794i)10-s + (−0.443 + 0.896i)12-s + (0.568 − 0.822i)13-s + (0.861 − 0.506i)14-s + (0.981 − 0.192i)15-s + (−0.168 + 0.985i)16-s + (0.443 + 0.896i)17-s + (−0.926 + 0.377i)18-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1441 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.708 + 0.706i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1441 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.708 + 0.706i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(5.688428107 + 2.351133463i\) |
\(L(\frac12)\) |
\(\approx\) |
\(5.688428107 + 2.351133463i\) |
\(L(1)\) |
\(\approx\) |
\(2.362881411 + 0.8371206817i\) |
\(L(1)\) |
\(\approx\) |
\(2.362881411 + 0.8371206817i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 11 | \( 1 \) |
| 131 | \( 1 \) |
good | 2 | \( 1 + (0.906 + 0.421i)T \) |
| 3 | \( 1 + (0.399 + 0.916i)T \) |
| 5 | \( 1 + (0.215 - 0.976i)T \) |
| 7 | \( 1 + (0.568 - 0.822i)T \) |
| 13 | \( 1 + (0.568 - 0.822i)T \) |
| 17 | \( 1 + (0.443 + 0.896i)T \) |
| 19 | \( 1 + (0.989 + 0.144i)T \) |
| 23 | \( 1 + (0.998 - 0.0483i)T \) |
| 29 | \( 1 + (-0.120 - 0.992i)T \) |
| 31 | \( 1 + (-0.836 + 0.548i)T \) |
| 37 | \( 1 + (0.970 + 0.239i)T \) |
| 41 | \( 1 + (0.715 + 0.698i)T \) |
| 43 | \( 1 + (0.215 - 0.976i)T \) |
| 47 | \( 1 + (0.681 - 0.732i)T \) |
| 53 | \( 1 + (0.309 - 0.951i)T \) |
| 59 | \( 1 + (-0.443 - 0.896i)T \) |
| 61 | \( 1 + (-0.809 - 0.587i)T \) |
| 67 | \( 1 + (-0.215 - 0.976i)T \) |
| 71 | \( 1 + (-0.779 - 0.626i)T \) |
| 73 | \( 1 + (0.809 + 0.587i)T \) |
| 79 | \( 1 + (0.861 + 0.506i)T \) |
| 83 | \( 1 + (-0.644 + 0.764i)T \) |
| 89 | \( 1 + (-0.809 + 0.587i)T \) |
| 97 | \( 1 + (0.943 - 0.331i)T \) |
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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
−20.60455326485619178874678612805, −19.672765982547720309722480816798, −18.85977390969553091407750707299, −18.42543809452934315314520617558, −17.91546052563765823080306085702, −16.53643812421850751468686790036, −15.5635567751483982196683572626, −14.75468609008538058349723039172, −14.2773927258361496169001379649, −13.71443283856169651593476663179, −12.87477345303078255742418290284, −12.00272220216500682872450977121, −11.39059980780680708711068500588, −10.89329800102144209788288516533, −9.48905716423894732796202713825, −8.989442382810807986548018249711, −7.42083478153459217610582107910, −7.23807107996351379895014563875, −6.026282897106735365400714359857, −5.64533383557581685544401977335, −4.41500963521746310077390704050, −3.15960997458762750539087812473, −2.76445715651086270376549456663, −1.83483152417821968513059010205, −1.03057451921507650010654959702,
0.892064489162832159427923228938, 2.04581515996316259015412831822, 3.347796625357790652180843940630, 3.87624740858219692440355176488, 4.78534348507781348503672506110, 5.326728837968198861725811260708, 6.10908170011023942107807936606, 7.55547691433727518966602855257, 8.04901917517613219476882580892, 8.81446588427600905494922718617, 9.84191691283522781455957292628, 10.744199857070253052136135117305, 11.37766930868410626360882339255, 12.45561923578790663997914599191, 13.25582099625217810544429453406, 13.79606860978174493257382350980, 14.58311443367637762723888482157, 15.31523500902711686559994012925, 15.976266385251934632465084047257, 16.92270123661646101773677620661, 16.99628249851102514552229774654, 18.119735011018896641926721011439, 19.7408384216288327728724067891, 20.10700951973484701003483749675, 20.97697735342156788158196635602