L(s) = 1 | + (−0.540 − 0.841i)3-s + (−0.654 − 0.755i)5-s + (0.755 − 0.654i)7-s + (−0.415 + 0.909i)9-s + (−0.654 + 0.755i)11-s + (−0.540 − 0.841i)13-s + (−0.281 + 0.959i)15-s + (0.959 − 0.281i)17-s + (0.909 + 0.415i)19-s + (−0.959 − 0.281i)21-s + (0.909 + 0.415i)23-s + (−0.142 + 0.989i)25-s + (0.989 − 0.142i)27-s + (−0.755 + 0.654i)29-s + (0.909 − 0.415i)31-s + ⋯ |
L(s) = 1 | + (−0.540 − 0.841i)3-s + (−0.654 − 0.755i)5-s + (0.755 − 0.654i)7-s + (−0.415 + 0.909i)9-s + (−0.654 + 0.755i)11-s + (−0.540 − 0.841i)13-s + (−0.281 + 0.959i)15-s + (0.959 − 0.281i)17-s + (0.909 + 0.415i)19-s + (−0.959 − 0.281i)21-s + (0.909 + 0.415i)23-s + (−0.142 + 0.989i)25-s + (0.989 − 0.142i)27-s + (−0.755 + 0.654i)29-s + (0.909 − 0.415i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 712 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.106 - 0.994i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 712 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.106 - 0.994i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.047409103 - 1.165207326i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.047409103 - 1.165207326i\) |
\(L(1)\) |
\(\approx\) |
\(0.8257962960 - 0.3977632288i\) |
\(L(1)\) |
\(\approx\) |
\(0.8257962960 - 0.3977632288i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 89 | \( 1 \) |
good | 3 | \( 1 + (-0.540 - 0.841i)T \) |
| 5 | \( 1 + (-0.654 - 0.755i)T \) |
| 7 | \( 1 + (0.755 - 0.654i)T \) |
| 11 | \( 1 + (-0.654 + 0.755i)T \) |
| 13 | \( 1 + (-0.540 - 0.841i)T \) |
| 17 | \( 1 + (0.959 - 0.281i)T \) |
| 19 | \( 1 + (0.909 + 0.415i)T \) |
| 23 | \( 1 + (0.909 + 0.415i)T \) |
| 29 | \( 1 + (-0.755 + 0.654i)T \) |
| 31 | \( 1 + (0.909 - 0.415i)T \) |
| 37 | \( 1 - iT \) |
| 41 | \( 1 + (0.540 - 0.841i)T \) |
| 43 | \( 1 + (0.755 + 0.654i)T \) |
| 47 | \( 1 + (0.841 + 0.540i)T \) |
| 53 | \( 1 + (0.841 - 0.540i)T \) |
| 59 | \( 1 + (-0.540 + 0.841i)T \) |
| 61 | \( 1 + (0.989 - 0.142i)T \) |
| 67 | \( 1 + (0.841 - 0.540i)T \) |
| 71 | \( 1 + (-0.654 + 0.755i)T \) |
| 73 | \( 1 + (0.415 + 0.909i)T \) |
| 79 | \( 1 + (0.415 + 0.909i)T \) |
| 83 | \( 1 + (0.281 + 0.959i)T \) |
| 97 | \( 1 + (-0.654 - 0.755i)T \) |
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
−22.479100533689091442570504342464, −21.78837009175534682536715056953, −21.194878451561825601462512493352, −20.41908628566709016898719470658, −19.06565624375940146995485201800, −18.68558528628286222484871744953, −17.69791836108752127466319557718, −16.79798124779252270032294430927, −15.98712971232630607972018982521, −15.221531960386410336408094908396, −14.650375343216974608974372195127, −13.77551510558508126352993487807, −12.225012994625476941631538689061, −11.651208977982808626472319911600, −11.020236721027714330536611432596, −10.20337200948937396153223751232, −9.198089062212415325442449072310, −8.24367180510940340233255703827, −7.31688272751265373645188918605, −6.18156929607437433332611999003, −5.281853128014346761883415355007, −4.496910788685308150813696267299, −3.36924050731562520230421083928, −2.552647921260022937275445093370, −0.775028343405058935771803851501,
0.62622293374494478481677040799, 1.2712263209870609302552023025, 2.61236141724406311733053632549, 4.00429603682973973774881221645, 5.2201372890305910760669457835, 5.41397049006672137568822692194, 7.32688023992539284277065648837, 7.47352019832562032801231275051, 8.25877768282933994219113361249, 9.60857876247307500232905291701, 10.65932795150712532266469701440, 11.44721450394926901471780600222, 12.35015628138594182654435532134, 12.7968533282480945020400134186, 13.78789569618230840535907429737, 14.72279184248139742015635437962, 15.74013789108085727251827569911, 16.63794441143208012476819575038, 17.34606124641344338620070677841, 17.97408467445961946112310481249, 18.90780312372895809354277652668, 19.781004755901931281367780833184, 20.50664480243965023193373461616, 21.10941219403762901144289323177, 22.65676029301083063952739358887