L(s) = 1 | + (−0.866 + 0.5i)3-s + (0.5 − 0.866i)9-s + (−0.5 − 0.866i)11-s − i·13-s + (0.866 − 0.5i)17-s + (−0.5 + 0.866i)19-s + (0.866 + 0.5i)23-s + i·27-s + 29-s + (−0.5 − 0.866i)31-s + (0.866 + 0.5i)33-s + (−0.866 − 0.5i)37-s + (−0.5 − 0.866i)39-s − 41-s + i·43-s + ⋯ |
L(s) = 1 | + (−0.866 + 0.5i)3-s + (0.5 − 0.866i)9-s + (−0.5 − 0.866i)11-s − i·13-s + (0.866 − 0.5i)17-s + (−0.5 + 0.866i)19-s + (0.866 + 0.5i)23-s + i·27-s + 29-s + (−0.5 − 0.866i)31-s + (0.866 + 0.5i)33-s + (−0.866 − 0.5i)37-s + (−0.5 − 0.866i)39-s − 41-s + i·43-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 280 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.910 + 0.413i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 280 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.910 + 0.413i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.1068181501 + 0.4929036306i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.1068181501 + 0.4929036306i\) |
\(L(1)\) |
\(\approx\) |
\(0.6889802340 + 0.1544277387i\) |
\(L(1)\) |
\(\approx\) |
\(0.6889802340 + 0.1544277387i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 \) |
good | 3 | \( 1 + (-0.866 + 0.5i)T \) |
| 11 | \( 1 + (-0.5 - 0.866i)T \) |
| 13 | \( 1 - iT \) |
| 17 | \( 1 + (0.866 - 0.5i)T \) |
| 19 | \( 1 + (-0.5 + 0.866i)T \) |
| 23 | \( 1 + (0.866 + 0.5i)T \) |
| 29 | \( 1 + T \) |
| 31 | \( 1 + (-0.5 - 0.866i)T \) |
| 37 | \( 1 + (-0.866 - 0.5i)T \) |
| 41 | \( 1 - T \) |
| 43 | \( 1 + iT \) |
| 47 | \( 1 + (0.866 + 0.5i)T \) |
| 53 | \( 1 + (-0.866 + 0.5i)T \) |
| 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 | \( 1 + (-0.866 + 0.5i)T \) |
| 79 | \( 1 + (-0.5 + 0.866i)T \) |
| 83 | \( 1 - iT \) |
| 89 | \( 1 + (-0.5 + 0.866i)T \) |
| 97 | \( 1 + iT \) |
show more | |
show less | |
\(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−25.090405249392831509467126580626, −23.83476277222804275890421225028, −23.2945871945280924681716452581, −22.456751745991947795057077670076, −21.51851744568476551432055680786, −20.45948661170048461559568185666, −19.391402977692269044199189176465, −18.4737846503174806024846923588, −17.593178290994521442661262891222, −16.98996453238912716048103359320, −15.77642531159776357008610348211, −14.96664517279997765444199981378, −13.56802843319872056393774307950, −12.66124499867112159387512438220, −12.07755377529366530985956405274, −10.675106069613032795050140914592, −10.244107815097218567275203650309, −8.61664832417528990021509443504, −7.506757201022014323194243630755, −6.66612125936894954433205600596, −5.44547079047866766427347533083, −4.68230888320562073635912097428, −2.96445274674087308301386672460, −1.55596407272335668305734840413, −0.18787731942642221200142912318,
1.28732340565882654189743057539, 3.12251848664993946698000014927, 4.29298816798447801553070571009, 5.395541311452557840241878863720, 6.25003374250426426870863217246, 7.43402913082105408129150908659, 8.78884588164817186881605422227, 9.8178988609538678155497607022, 10.77545002329182342950367723893, 11.61236789118757472385037007891, 12.496314376324470876069382302825, 13.728334901940497464928112315124, 14.74879435475674686246107132918, 15.91235754018163342072143911835, 16.53648209279976973836056998845, 17.33760040622384528729245435303, 18.55842627233528209845673727892, 19.111802267500738684738498589587, 20.735245420603494226914256527934, 21.27464326202443012929432107890, 22.09128647859151496114972743554, 23.24091886378836211168504048658, 23.65901337433049901212115376986, 24.78851097235565389032480656443, 25.92673447921288610076438590307