L(s) = 1 | + i·7-s + 11-s + (−0.866 − 0.5i)13-s + (−0.866 + 0.5i)17-s + (−0.866 − 0.5i)23-s + (0.5 − 0.866i)29-s + 31-s − i·37-s + (0.5 + 0.866i)41-s + (0.866 − 0.5i)43-s + (0.866 + 0.5i)47-s − 49-s + (0.866 + 0.5i)53-s + (0.5 + 0.866i)59-s + (0.5 − 0.866i)61-s + ⋯ |
L(s) = 1 | + i·7-s + 11-s + (−0.866 − 0.5i)13-s + (−0.866 + 0.5i)17-s + (−0.866 − 0.5i)23-s + (0.5 − 0.866i)29-s + 31-s − i·37-s + (0.5 + 0.866i)41-s + (0.866 − 0.5i)43-s + (0.866 + 0.5i)47-s − 49-s + (0.866 + 0.5i)53-s + (0.5 + 0.866i)59-s + (0.5 − 0.866i)61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2280 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.720 + 0.693i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2280 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.720 + 0.693i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.427443060 + 0.5754651308i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.427443060 + 0.5754651308i\) |
\(L(1)\) |
\(\approx\) |
\(1.066271680 + 0.1434850526i\) |
\(L(1)\) |
\(\approx\) |
\(1.066271680 + 0.1434850526i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| 19 | \( 1 \) |
good | 7 | \( 1 + iT \) |
| 11 | \( 1 + T \) |
| 13 | \( 1 + (-0.866 - 0.5i)T \) |
| 17 | \( 1 + (-0.866 + 0.5i)T \) |
| 23 | \( 1 + (-0.866 - 0.5i)T \) |
| 29 | \( 1 + (0.5 - 0.866i)T \) |
| 31 | \( 1 + T \) |
| 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 | \( 1 + (0.866 + 0.5i)T \) |
| 59 | \( 1 + (0.5 + 0.866i)T \) |
| 61 | \( 1 + (0.5 - 0.866i)T \) |
| 67 | \( 1 + (0.866 + 0.5i)T \) |
| 71 | \( 1 + (0.5 + 0.866i)T \) |
| 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 | \( 1 + (0.866 - 0.5i)T \) |
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
−19.63841361420714492717280099273, −19.071454196373969351498360420365, −17.90418707331365453249726964525, −17.43927206768095949286833069312, −16.7507669863492100128374287531, −16.090398253591936912812391029209, −15.248055835972973096055514037613, −14.271827450610113930497130438585, −13.966520732110219864635525454536, −13.1630668785746997155648985454, −12.131864833507508063574051139118, −11.65737351116132665377606097058, −10.7730099280380433959510297389, −9.9917792905917087454348788001, −9.34821918940478544146063429016, −8.52990584065318319840891166101, −7.54233095293518614115922935336, −6.89615355491607261450542860009, −6.32724267862590043293463670571, −5.10359035830098171649958738478, −4.344084081450026466536653126477, −3.755335607160448357848008238900, −2.63267712280081402368463662706, −1.67430872138654735213151928929, −0.64626568632487247842773391376,
0.887021646580050001612430609186, 2.22925144209588323551591474839, 2.59808459509913022748116473197, 3.92984879105178537309506909354, 4.53137646746613580734841025137, 5.6251081583330785932254182217, 6.19472033054415383576516048274, 7.01271380228158784341695169650, 8.03582344766705855119983290500, 8.672094972328719340222948457539, 9.43311512017994203437327262763, 10.11339563986725149768690223178, 11.06223847326687043858522003034, 11.90527580919340822088731829780, 12.334356530835446192023113507195, 13.11948563957293936753562120032, 14.15156493024550797906449426960, 14.66978626812559252284490831569, 15.486423946107479936534216323, 15.965629038313576127218741872269, 17.091951077097472058031347551235, 17.52372382438612925539932252868, 18.25766591789829726204613914163, 19.199476482049919927388232150484, 19.60685326995681841026257374766