L(s) = 1 | + (0.841 − 0.540i)2-s + (−0.936 − 0.349i)3-s + (0.415 − 0.909i)4-s + (0.989 − 0.142i)5-s + (−0.977 + 0.212i)6-s + (0.800 + 0.599i)7-s + (−0.142 − 0.989i)8-s + (0.755 + 0.654i)9-s + (0.755 − 0.654i)10-s + (0.142 − 0.989i)11-s + (−0.707 + 0.707i)12-s + (0.349 − 0.936i)13-s + (0.997 + 0.0713i)14-s + (−0.977 − 0.212i)15-s + (−0.654 − 0.755i)16-s + (−0.540 + 0.841i)17-s + ⋯ |
L(s) = 1 | + (0.841 − 0.540i)2-s + (−0.936 − 0.349i)3-s + (0.415 − 0.909i)4-s + (0.989 − 0.142i)5-s + (−0.977 + 0.212i)6-s + (0.800 + 0.599i)7-s + (−0.142 − 0.989i)8-s + (0.755 + 0.654i)9-s + (0.755 − 0.654i)10-s + (0.142 − 0.989i)11-s + (−0.707 + 0.707i)12-s + (0.349 − 0.936i)13-s + (0.997 + 0.0713i)14-s + (−0.977 − 0.212i)15-s + (−0.654 − 0.755i)16-s + (−0.540 + 0.841i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 89 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.163 - 0.986i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 89 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.163 - 0.986i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.632052923 - 1.924607927i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.632052923 - 1.924607927i\) |
\(L(1)\) |
\(\approx\) |
\(1.400302635 - 0.8645716883i\) |
\(L(1)\) |
\(\approx\) |
\(1.400302635 - 0.8645716883i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 89 | \( 1 \) |
good | 2 | \( 1 + (0.841 - 0.540i)T \) |
| 3 | \( 1 + (-0.936 - 0.349i)T \) |
| 5 | \( 1 + (0.989 - 0.142i)T \) |
| 7 | \( 1 + (0.800 + 0.599i)T \) |
| 11 | \( 1 + (0.142 - 0.989i)T \) |
| 13 | \( 1 + (0.349 - 0.936i)T \) |
| 17 | \( 1 + (-0.540 + 0.841i)T \) |
| 19 | \( 1 + (-0.997 + 0.0713i)T \) |
| 23 | \( 1 + (-0.0713 - 0.997i)T \) |
| 29 | \( 1 + (0.599 - 0.800i)T \) |
| 31 | \( 1 + (-0.0713 + 0.997i)T \) |
| 37 | \( 1 + (0.707 + 0.707i)T \) |
| 41 | \( 1 + (-0.349 - 0.936i)T \) |
| 43 | \( 1 + (0.599 + 0.800i)T \) |
| 47 | \( 1 + (0.909 + 0.415i)T \) |
| 53 | \( 1 + (-0.909 + 0.415i)T \) |
| 59 | \( 1 + (0.936 - 0.349i)T \) |
| 61 | \( 1 + (-0.877 + 0.479i)T \) |
| 67 | \( 1 + (0.415 + 0.909i)T \) |
| 71 | \( 1 + (-0.989 - 0.142i)T \) |
| 73 | \( 1 + (0.654 + 0.755i)T \) |
| 79 | \( 1 + (-0.755 + 0.654i)T \) |
| 83 | \( 1 + (0.977 - 0.212i)T \) |
| 97 | \( 1 + (-0.142 - 0.989i)T \) |
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
−30.44153966019217743444319789441, −29.66245041479555221890070337941, −28.67268871732871141550989439460, −27.33148683879154673396570433273, −26.18697525350825561497805120725, −25.15993144952034067841107391159, −23.914192952474736878928122375060, −23.219170979616083861436900953561, −22.07442428795214943674749239007, −21.29213831863554455629512567671, −20.44893085454755116275681546525, −18.07522755126544244721314697450, −17.37232979597194212625601794914, −16.5420999438507792422860726656, −15.20081356072482850528515696108, −14.13709087445930071966676277017, −13.06484310713018133085536132008, −11.72847228762607887768527151361, −10.747220341409402763948321211301, −9.26408543754343602269747708249, −7.25363122556603709747818336571, −6.346227245472486322075808610768, −5.03934324113325102311851605307, −4.190703017321729371625219501638, −1.90466508637962537169439089303,
1.10614791907461548109146164094, 2.41945737059094754278144158183, 4.55515679102469011321868072686, 5.74650480135223723527425116219, 6.32213544813320090785008900571, 8.51813645986015911852059613554, 10.39145197327910865471220155344, 11.052553660118636463044051882146, 12.37617806634154603250238709480, 13.192419362411508385028908670517, 14.34578360415605328154404347471, 15.672257424454742334526851940197, 17.11461331657367218646355439521, 18.13067905261311024724441593555, 19.14714772405507524877819073311, 20.78529292349543943033843415312, 21.64548431744707038905473485964, 22.269804872066729268948196786609, 23.60431548159787727377820669087, 24.47044391421815548112564184330, 25.19775163671368540737049913128, 27.29189757504709097469533600900, 28.31085658975654358692523446528, 28.98890517634938304713168208879, 30.05520993158400485018993286667