500 Riemann Zeros · Shell-Residue Decomposition · mpmath dps=30
γₙ = Sₙ − rₙ
n = 1 … 500 · Sₙ = (kₙ+½)π · rₙ = Sₙ−γₙ · uₙ = 2|rₙ|/π ∈ [0,1)
FLAGSHIP |r|<0.01 DEEP |r|<0.05 STRONG |r|<0.2 FLOOR r>0 CEILING r<0
Flagship zeros |r|<0.01
2
n=1 (r=0.00244) · n=438 (r=0.00381) · gap = 437
γ₁ rank among n≤500
#1
|r₁| = 0.00244180 · closest of all 500
Deep alignment |r|<0.05
16
3.2% of 500 zeros — including n=45,150,251,280,340,342…
Strong floor |r|<0.2
SIGN DISTRIBUTION
253 floor / 247 ceiling
r>0 (below shell): 50.6% · r<0 (above shell): 49.4%
ALTERNATION RATE
68.5%
consecutive pairs alternate sign · 157/499 same-sign (31.5%)
u-Distribution Histogram · n=1..500
Shared Shells · 3+ Zeros per Shell · Top 24
Triple Shells — 56 instances in n≤500

A triple shell means 3 zeros cluster in one π-interval. This is directly linked to Gram's law violations. The bracket structure (one Floor, one Ceiling within a shared shell) is strongly dominant — 81.8% of pairs.

k=64 · S=202.6327 · 3 zeros
γ=201.2648 r=+1.3680 FLOOR
γ=202.4936 r=+0.1391 STRONG
γ=204.1897 r=-1.5569 CEILING
k=73 · S=230.9071 · 3 zeros
γ=229.3374 r=+1.5696 FLOOR
γ=231.2502 r=-0.3431 CEILING
γ=231.9872 r=-1.0802 CEILING
k=87 · S=274.8894 · 3 zeros
γ=273.4596 r=+1.4297 FLOOR
γ=275.5875 r=-0.6981 CEILING
γ=276.4520 r=-1.5627 CEILING
k=97 · S=306.3053 · 3 zeros
γ=304.8644 r=+1.4409 FLOOR
γ=305.7289 r=+0.5764 FLOOR
γ=307.2195 r=-0.9142 CEILING
k=99 · S=312.5885 · 3 zeros
γ=311.1651 r=+1.4233 FLOOR
γ=312.4278 r=+0.1607 STRONG
γ=313.9853 r=-1.3968 CEILING
k=102 · S=322.0132 · 3 zeros
γ=321.1601 r=+0.8531 FLOOR
γ=322.1446 r=-0.1313 STRONG
γ=323.4670 r=-1.4537 CEILING
k=108 · S=340.8628 · 3 zeros
γ=339.8582 r=+1.0046 FLOOR
γ=341.0423 r=-0.1795 STRONG
γ=342.0549 r=-1.1921 CEILING
k=113 · S=356.5708 · 3 zeros
γ=356.0176 r=+0.5532 FLOOR
γ=357.1513 r=-0.5805 CEILING
γ=357.9527 r=-1.3819 CEILING
k=117 · S=369.1371 · 3 zeros
γ=367.9936 r=+1.1436 FLOOR
γ=368.9684 r=+0.1687 STRONG
γ=370.0509 r=-0.9138 CEILING
k=119 · S=375.4203 · 3 zeros
γ=373.8649 r=+1.5554 FLOOR
γ=375.8259 r=-0.4056 CEILING
γ=376.3241 r=-0.9038 CEILING
k=122 · S=384.8451 · 3 zeros
γ=383.4435 r=+1.4016 FLOOR
γ=384.9561 r=-0.1110 STRONG
γ=385.8613 r=-1.0162 CEILING
k=128 · S=403.6947 · 3 zeros
γ=402.8619 r=+0.8327 FLOOR
γ=404.2364 r=-0.5418 CEILING
γ=405.1344 r=-1.4397 CEILING
k=133 · S=419.4026 · 3 zeros
γ=418.3877 r=+1.0149 FLOOR
γ=419.8614 r=-0.4587 CEILING
γ=420.6438 r=-1.2412 CEILING
k=140 · S=441.3938 · 3 zeros
γ=439.9184 r=+1.4753 FLOOR
γ=441.6832 r=-0.2894 CEILING
γ=442.9045 r=-1.5108 CEILING
k=142 · S=447.6770 · 3 zeros
γ=446.8606 r=+0.8163 FLOOR
γ=447.4417 r=+0.2352 FLOOR
γ=449.1485 r=-1.4716 CEILING
k=148 · S=466.5265 · 3 zeros
γ=465.6715 r=+0.8550 FLOOR
γ=466.5703 r=-0.0438 DEEP
γ=467.4390 r=-0.9125 CEILING
k=154 · S=485.3761 · 3 zeros
γ=483.8514 r=+1.5246 FLOOR
γ=485.5391 r=-0.1631 STRONG
γ=486.5287 r=-1.1527 CEILING
k=157 · S=494.8008 · 3 zeros
γ=493.3144 r=+1.4864 FLOOR
γ=493.9580 r=+0.8428 FLOOR
γ=495.3588 r=-0.5580 CEILING
k=159 · S=501.0840 · 3 zeros
γ=500.3091 r=+0.7749 FLOOR
γ=501.6044 r=-0.5204 CEILING
γ=502.2763 r=-1.1922 CEILING
k=164 · S=516.7920 · 3 zeros
γ=515.4351 r=+1.3569 FLOOR
γ=517.5897 r=-0.7977 CEILING
γ=518.2342 r=-1.4422 CEILING
k=166 · S=523.0752 · 3 zeros
γ=521.5252 r=+1.5500 FLOOR
γ=522.4567 r=+0.6185 FLOOR
γ=523.9605 r=-0.8854 CEILING
k=168 · S=529.3584 · 4 zeros
γ=527.9036 r=+1.4547 FLOOR
γ=528.4062 r=+0.9521 FLOOR
γ=529.8062 r=-0.4479 CEILING
γ=530.8669 r=-1.5086 CEILING
k=174 · S=548.2079 · 3 zeros
γ=547.0109 r=+1.1970 FLOOR
γ=547.9316 r=+0.2763 FLOOR
γ=549.4976 r=-1.2896 CEILING
k=179 · S=563.9159 · 3 zeros
γ=562.5592 r=+1.3567 FLOOR
γ=564.1609 r=-0.2450 CEILING
γ=564.5061 r=-0.5902 CEILING
Tightest Bracket Pairs (sum r₁+r₂ closest to 0)
n pairkγ valuesr₁r₂sum r₁+r₂sign
326,327k=184579.0988, 580.1370+0.5250-0.5131Σ=+0.0119⇄ opposite
240,241k=145456.3284, 457.9039+0.7733-0.8022Σ=-0.0289⇄ opposite
104,105k=77242.8233, 244.0709+0.6502-0.5975Σ=+0.0527⇄ opposite
102,103k=76239.5555, 241.0492+0.7764-0.7173Σ=+0.0590⇄ opposite
488,489k=253795.6066, 797.2635+0.7871-0.8697Σ=-0.0826⇄ opposite
43,44k=41129.5787, 131.0877+0.7974-0.7116Σ=+0.0858⇄ opposite
100,101k=75236.5242, 237.7698+0.6660-0.5796Σ=+0.0864⇄ opposite
194,195k=123387.2229, 388.8461+0.7638-0.8594Σ=-0.0956⇄ opposite
236,237k=143450.1269, 451.4033+0.6916-0.5848Σ=+0.1068⇄ opposite
155,156k=104327.4439, 329.0331+0.8525-0.7366Σ=+0.1159⇄ opposite
Deserted Shells (Gram Law Failures) · k = 5, 8, 12, 14, 23
k=5   S=17.2788   gap: zeros jump from k=4 to k=6
k=8   S=26.7035   gap: zeros jump from k=7 to k=9
k=12   S=39.2699   gap: zeros jump from k=11 to k=13
k=14   S=45.5531   gap: zeros jump from k=13 to k=15
k=23   S=73.8274   gap: zeros jump from k=22 to k=24
Full 500-Zero Table · mpmath dps=30
γₙ = Sₙ − rₙ · Complete Record
500 rows
nγₙkₙSₙ=(kₙ+½)π rₙ=Sₙ−γₙ|rₙ|u=2|r|/πρₙ=γₙ/Sₙtag