Generating a scatter plot in Matplotlib with negative and positive axes

I am working on a project that plots clinical values using Matplotlib and want to display a y-axis with both negative and positive values going from -3 to 3. I'm getting the data from a DataFrame.

An example of the data I'm trying to plot:

analyte_name = ['Uric Acid - Basic', 'Urea', 'Triglycerides - Basic', 'Sodium', 'Potassium - Basic', 'Glucose - Basic', 'Gamma Glutamytranferase - Basic', 'Creatinine - Basic', 'Cholesterol Total - Basic', 'Cholesterol LDL - Basic', 'Cholesterol HDL - Basic', 'Chloride - Basic']
z_scores = ['-0.10', '-0.60', '-0.01', '-0.77', '-12.95', '-0.55', '-0.58', '-0.37', '-0.07', '0.19', '0.88', '0.69']

This is what I could come up with:

import matplotlib.pyplot as plt
import pandas as pd
import seaborn as sns
import numpy as np

df = pd.DataFrame({'x_':analyte_names, 'y_':z_scores})
fig = plt.figure()
ax = fig.add_subplot(111)

ax.set_xlabel('analyte name')
ax.set_ylabel('z-score')

# plt.axhline(0, color='black')
plt.ylim(-3, 3)
plt.xticks(rotation=90)
plt.scatter('x_', 'y_' ,data=df, marker='o')
# plt.style.use('seaborn-dark')
plt.show()

But this gives me a plot that looks like this:

y-axis plotted in sequence from z_scores[0] onwards but not displaying all z_scores

Commenting out the plt.ylim(-3, 3) line gives me an image like this:

y-axis plotted in sequence from z_scores[0] onwards and displaying all z_score but in sequence

The code I'm using is modified from one I tried using before which was:

df = pd.DataFrame({'x_':['A','B','C','D','E'], 
'y_':np.random.uniform(-3,3,5)})

fig = plt.figure()
ax = fig.add_subplot(111)

# ax.spines['top'].set_visible(False)
# ax.spines['right'].set_visible(False)

ax.set_xlabel('sample')
ax.set_ylabel('z-score')

plt.axhline(0, color='black')
plt.ylim(-3, 3)
plt.scatter('x_', 'y_' ,data=df, marker='o')
# plt.style.use('seaborn-dark')
plt.show()

That code generated what I want my final output to look like before some slight styling:

y axis with negative an positive values

I've been trying to use different methods to pass the data to the x and y axis like passing it as a dictionary but the results have been the same.

I'm still learning how to plot data and hope to can get some help.

Thanks.