L(s) = 1 | + (0.342 + 0.939i)2-s + (−0.984 − 0.173i)3-s + (−0.766 + 0.642i)4-s + (−0.173 − 0.984i)6-s + (−0.866 + 0.5i)7-s + (−0.866 − 0.5i)8-s + (0.939 + 0.342i)9-s + (0.866 − 0.5i)12-s + (−0.984 + 0.173i)13-s + (−0.766 − 0.642i)14-s + (0.173 − 0.984i)16-s + (0.342 + 0.939i)17-s + i·18-s + (0.939 − 0.342i)21-s + (−0.642 − 0.766i)23-s + (0.766 + 0.642i)24-s + ⋯ |
L(s) = 1 | + (0.342 + 0.939i)2-s + (−0.984 − 0.173i)3-s + (−0.766 + 0.642i)4-s + (−0.173 − 0.984i)6-s + (−0.866 + 0.5i)7-s + (−0.866 − 0.5i)8-s + (0.939 + 0.342i)9-s + (0.866 − 0.5i)12-s + (−0.984 + 0.173i)13-s + (−0.766 − 0.642i)14-s + (0.173 − 0.984i)16-s + (0.342 + 0.939i)17-s + i·18-s + (0.939 − 0.342i)21-s + (−0.642 − 0.766i)23-s + (0.766 + 0.642i)24-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.991 - 0.128i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.991 - 0.128i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.4245938352 + 0.02737141461i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.4245938352 + 0.02737141461i\) |
\(L(1)\) |
\(\approx\) |
\(0.5541360317 + 0.2958053446i\) |
\(L(1)\) |
\(\approx\) |
\(0.5541360317 + 0.2958053446i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 \) |
| 11 | \( 1 \) |
| 19 | \( 1 \) |
good | 2 | \( 1 + (0.342 + 0.939i)T \) |
| 3 | \( 1 + (-0.984 - 0.173i)T \) |
| 7 | \( 1 + (-0.866 + 0.5i)T \) |
| 13 | \( 1 + (-0.984 + 0.173i)T \) |
| 17 | \( 1 + (0.342 + 0.939i)T \) |
| 23 | \( 1 + (-0.642 - 0.766i)T \) |
| 29 | \( 1 + (-0.939 - 0.342i)T \) |
| 31 | \( 1 + (-0.5 - 0.866i)T \) |
| 37 | \( 1 + iT \) |
| 41 | \( 1 + (-0.173 + 0.984i)T \) |
| 43 | \( 1 + (-0.642 + 0.766i)T \) |
| 47 | \( 1 + (0.342 - 0.939i)T \) |
| 53 | \( 1 + (-0.642 - 0.766i)T \) |
| 59 | \( 1 + (0.939 - 0.342i)T \) |
| 61 | \( 1 + (-0.766 + 0.642i)T \) |
| 67 | \( 1 + (0.342 - 0.939i)T \) |
| 71 | \( 1 + (0.766 + 0.642i)T \) |
| 73 | \( 1 + (0.984 + 0.173i)T \) |
| 79 | \( 1 + (0.173 - 0.984i)T \) |
| 83 | \( 1 + (0.866 - 0.5i)T \) |
| 89 | \( 1 + (-0.173 - 0.984i)T \) |
| 97 | \( 1 + (-0.342 - 0.939i)T \) |
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
−21.80028186787236756628038784406, −20.80453211330926884670751662738, −20.06354445550664571742956096393, −19.30959975916698872049507256183, −18.5283546630820820339150158846, −17.730404980914091638311849489819, −16.988041743158221543690098925329, −16.133996292621316157237164245774, −15.32977752422505488679543197854, −14.236707859613517502508164352111, −13.491568695411156549636745245919, −12.458429411108007403151718728215, −12.236188159531214448003852542650, −11.14873971055363531851239578020, −10.51874984729461697040943713188, −9.68175356022955235516729918053, −9.26652913359664669724587480515, −7.56316306561789272137639825580, −6.77226182458267756390149283640, −5.63394682256129594967172911244, −5.11330994660406268735538021419, −4.027245410134489192000912832443, −3.324528808075951556368609741348, −2.09537410689107396337642369702, −0.82423331965590701475206464249,
0.24810914599732788293584023525, 2.064242866946971443228883950309, 3.41254697827587930729680126154, 4.3972279551967785027958510113, 5.264986307130348146518979499091, 6.08389747661380257584234357032, 6.58315647122193924157687534150, 7.51811080630088473083128364028, 8.36773827377559635671675650002, 9.61109046164953153907683901232, 10.05400380432086693453202385959, 11.44028299450200333227062616356, 12.250572503049380384142637563789, 12.80388217773984527763580510811, 13.47532536408564213586775113671, 14.751689727875627096681213842907, 15.23139141862768238022684514054, 16.267361098626749748454147698539, 16.73014595299838329630985047403, 17.324489131597706164923327828900, 18.39770150407224314935772397127, 18.78597379163569386522355518620, 19.8210877271305074944623404376, 21.18081126995413949567873012286, 21.97928049180247558886075911563