## Zeta zero approximations

$7 \pi -\text{Log}\left[\frac{7}{2} e^{-7 \pi /2}+\frac{5}{2} e^{-5 \pi /2}+\frac{3}{2} e^{-3 \pi /2}+e^{5 \pi /2}+2 \pi \right]$

$9 \pi -\text{Log}\left[-3 e^{\pi /2}-3 e^{2 \pi /2}-e^{3 \pi /2}+3 e^{4 \pi /2 }\right]$

$11 \pi -\text{Log}\left[-1+3 e^{4 \pi /2}+e^{6 \pi /2}\right]$

$13 \pi -\text{Log}\left[2-e^{\pi /2}+3 e^{2 \pi /2}+2 e^{3 \pi /2}+2 e^{4 \pi /2}-2 e^{5 \pi /2}+3 e^{6 \pi /2}\right]$

$15 \pi -\text{Log}\left[1-e^{\pi /2}+e^{2 \pi /2}-4 e^{3 \pi /2}+2 e^{4 \pi /2}+5 e^{5 \pi /2}+e^{7 \pi /2}+e^{9 \pi /2}\right]$

In[447]:= N[
7*Pi – Log[
2*Pi + Exp[5/2*Pi] + 3/2*Exp[-3/2*Pi] + 5/2*Exp[-5/2*Pi] +
7/2*Exp[-7/2*Pi]], 90]
7*Pi – Log[
2*Pi + Exp[5/2*Pi] + 3/2*Exp[-3/2*Pi] + 5/2*Exp[-5/2*Pi] +
7/2*Exp[-7/2*Pi]]
N[9*Pi – Log[
Exp[4/2*Pi]*3 – Exp[3/2*Pi] – Exp[2/2*Pi]*3 – Exp[1/2*Pi]*3], 90]
9*Pi – Log[Exp[4/2*Pi]*3 – Exp[3/2*Pi] – Exp[2/2*Pi]*3 – Exp[1/2*Pi]*3]
N[11*Pi – Log[Exp[6/2*Pi] + Exp[4/2*Pi]*3 – 1], 90]
11*Pi – Log[Exp[6/2*Pi] + Exp[4/2*Pi]*3 – 1]
N[13*Pi –
Log[Exp[6/2*Pi]*3 – Exp[5/2*Pi]*2 + Exp[4/2*Pi]*2 + Exp[3/2*Pi]*2 +
Exp[2/2*Pi]*3 – Exp[1/2*Pi] + 2], 90]
13*Pi – Log[
Exp[6/2*Pi]*3 – Exp[5/2*Pi]*2 + Exp[4/2*Pi]*2 + Exp[3/2*Pi]*2 +
Exp[2/2*Pi]*3 – Exp[1/2*Pi] + 2]
N[15*Pi –
Log[Exp[9/2*Pi] + Exp[7/2*Pi] + Exp[5/2*Pi]*5 + Exp[4/2*Pi]*2 –
Exp[3/2*Pi]*4 + Exp[2/2*Pi] – Exp[1/2*Pi] + Exp[0/2*Pi]], 90]
15*Pi – Log[
Exp[9/2*Pi] + Exp[7/2*Pi] + Exp[5/2*Pi]*5 + Exp[4/2*Pi]*2 –
Exp[3/2*Pi]*4 + Exp[2/2*Pi] – Exp[1/2*Pi] + Exp[0/2*Pi]]

Out[447]= \
14.1347251415462971625332949457130250888808428761125331718801906227734\
522626031127266673111

Out[448]=
7 \[Pi] –
Log[7/2 E^(-7 \[Pi]/2) + 5/2 E^(-5 \[Pi]/2) + 3/2 E^(-3 \[Pi]/2) +
E^(5 \[Pi]/2) + 2 \[Pi]]

Out[449]= \
21.0220647317531170031433976766645381602165975607485034136361666286850\
112342614339440360907

Out[450]=
9 \[Pi] –
Log[-3 E^(\[Pi]/2) – 3 E^\[Pi] – E^(3 \[Pi]/2) + 3 E^(2 \[Pi])]

Out[451]= \
25.0109121181194454425895620012384712403356051145851908039782928528267\
355833273049906375471

Out[452]= 11 \[Pi] – Log[-1 + 3 E^(2 \[Pi]) + E^(3 \[Pi])]

Out[453]= \
30.4248954527601648230070306069243251177298536494717015917080755061626\
004025280687729937055

Out[454]=
13 \[Pi] –
Log[2 – E^(\[Pi]/2) + 3 E^\[Pi] + 2 E^(3 \[Pi]/2) + 2 E^(2 \[Pi]) –
2 E^(5 \[Pi]/2) + 3 E^(3 \[Pi])]

Out[455]= \
32.9350618199689987097953015374911208470972884058585555099783653166032\
622776454859421331614

Out[456]=
15 \[Pi] –
Log[1 – E^(\[Pi]/2) + E^\[Pi] – 4 E^(3 \[Pi]/2) + 2 E^(2 \[Pi]) +
5 E^(5 \[Pi]/2) + E^(7 \[Pi]/2) + E^(9 \[Pi]/2)]