L(s) = 1 | + (−0.0724 − 0.997i)2-s + (0.485 + 0.873i)3-s + (−0.989 + 0.144i)4-s + (0.981 − 0.192i)5-s + (0.836 − 0.548i)6-s + (−0.779 + 0.626i)7-s + (0.215 + 0.976i)8-s + (−0.527 + 0.849i)9-s + (−0.262 − 0.964i)10-s + (−0.607 − 0.794i)12-s + (0.262 − 0.964i)13-s + (0.681 + 0.732i)14-s + (0.644 + 0.764i)15-s + (0.958 − 0.285i)16-s + (0.0241 − 0.999i)17-s + (0.885 + 0.464i)18-s + ⋯ |
L(s) = 1 | + (−0.0724 − 0.997i)2-s + (0.485 + 0.873i)3-s + (−0.989 + 0.144i)4-s + (0.981 − 0.192i)5-s + (0.836 − 0.548i)6-s + (−0.779 + 0.626i)7-s + (0.215 + 0.976i)8-s + (−0.527 + 0.849i)9-s + (−0.262 − 0.964i)10-s + (−0.607 − 0.794i)12-s + (0.262 − 0.964i)13-s + (0.681 + 0.732i)14-s + (0.644 + 0.764i)15-s + (0.958 − 0.285i)16-s + (0.0241 − 0.999i)17-s + (0.885 + 0.464i)18-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1441 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.412 - 0.910i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1441 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.412 - 0.910i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.367969764 - 0.8821199147i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.367969764 - 0.8821199147i\) |
\(L(1)\) |
\(\approx\) |
\(1.123678748 - 0.3324536219i\) |
\(L(1)\) |
\(\approx\) |
\(1.123678748 - 0.3324536219i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 11 | \( 1 \) |
| 131 | \( 1 \) |
good | 2 | \( 1 + (-0.0724 - 0.997i)T \) |
| 3 | \( 1 + (0.485 + 0.873i)T \) |
| 5 | \( 1 + (0.981 - 0.192i)T \) |
| 7 | \( 1 + (-0.779 + 0.626i)T \) |
| 13 | \( 1 + (0.262 - 0.964i)T \) |
| 17 | \( 1 + (0.0241 - 0.999i)T \) |
| 19 | \( 1 + (0.568 - 0.822i)T \) |
| 23 | \( 1 + (-0.995 + 0.0965i)T \) |
| 29 | \( 1 + (-0.527 - 0.849i)T \) |
| 31 | \( 1 + (0.861 - 0.506i)T \) |
| 37 | \( 1 + (0.168 - 0.985i)T \) |
| 41 | \( 1 + (-0.958 - 0.285i)T \) |
| 43 | \( 1 + (0.906 + 0.421i)T \) |
| 47 | \( 1 + (0.970 + 0.239i)T \) |
| 53 | \( 1 + (0.309 + 0.951i)T \) |
| 59 | \( 1 + (-0.943 - 0.331i)T \) |
| 61 | \( 1 + (0.809 + 0.587i)T \) |
| 67 | \( 1 + (0.906 - 0.421i)T \) |
| 71 | \( 1 + (0.748 + 0.663i)T \) |
| 73 | \( 1 + (-0.809 + 0.587i)T \) |
| 79 | \( 1 + (0.120 - 0.992i)T \) |
| 83 | \( 1 + (0.715 + 0.698i)T \) |
| 89 | \( 1 + (0.309 - 0.951i)T \) |
| 97 | \( 1 + (0.998 - 0.0483i)T \) |
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
−20.81174856752637312543434009112, −19.954655488103015034215481425395, −19.02351442413996067714835965505, −18.59363152656564889309970476056, −17.824062732303195189222845132666, −17.03508500779720344247702223308, −16.563525307785247734346378414130, −15.567310161707705670959377021688, −14.512238728392877188459206404739, −14.042420783139406975005191431631, −13.50864564386699114991024357850, −12.823857696227007287539681042, −12.023854604203456107001134399450, −10.50201526553549210428455243002, −9.833689616566692133977842048082, −9.09921309472197995876984507159, −8.31483577312132610494890676861, −7.44612813800694157108099737196, −6.52506480210010497831428521666, −6.35047677884095661262272898755, −5.37605725470036079411197118522, −4.0096697625305128649242315727, −3.30063639005227276624090943802, −1.9318830964975346970456760686, −1.10195858304168734275734642791,
0.67260543158870344534147202917, 2.2568745420768431534873067674, 2.66419144927045048131886074846, 3.504063927245758447951568246200, 4.53224037340992048242023300726, 5.438205577582778119143383729037, 5.94614148302377425953112248327, 7.57886651204092668972538274461, 8.592251680525172695858800187064, 9.28817843170818752201398597284, 9.73438879655714984144268178758, 10.34672836868438062782640665710, 11.250930729774593101153569621962, 12.1438946464409403474517761982, 13.077511234786278100713422824536, 13.63623371935463177034812562875, 14.23523896020113532048936117547, 15.37493911824666738970495854671, 15.94994089568810121440457041418, 16.95696330358134274641634570893, 17.70336142106012203611042615712, 18.44531817153841068807181648965, 19.22168865148193947843709287542, 20.20704527753244178051169430849, 20.4378716308138600279812115005