Story of Alakshmi is described in detail in Chapter 6 Uttarabhaga of Linga Mahapurana.
Lord Vishnu created Alakshmi as a part of His divine sports. Since she was created before Goddess lakshmi, she is called Jyestha. But she came into world during churning of ocean.
अनादिनिधनः श्रीमान्धाता नारायणः प्रभुः॥२॥ जगद्द्वैधमिदं
चक्रे मोहनाय जगत्पति:। विष्णुर्वे ब्राह्मणान्वेदान्वेदधमनि सनातनान्
॥३॥ श्रियं पद्मां तथा श्रेष्ठां भागमेकमकारयत्। ज्येष्ठामलक्ष्मीमशुभां
वेदबाह्यान्नराधमान्॥४॥ अधर्म च महातेजा भागमेकमकल्पयत्।
अलक्ष्मीमग्रतः सृष्ट्वा पश्चात्पद्मां जनार्दनः॥५॥ ज्येष्ठा तेन समाख्याता अलक्ष्मीद्वैिजसत्तमाः । अमृतोद्रववेलायां
विषानंतरमुल्बणात्॥६॥ अशुभा सा तथोत्पन्ना ज्येष्ठा इति च वै श्रुतम्।
ततः श्रीश्च समुत्पन्ना पद्मा विष्णुपरिग्रहः ॥७॥
In order to delude the universe, the glorious lord
Visnu, who is without beginning or the end and is the lord of the
universe, created the two-fold aspects of the universe. Visnu created
one set consisting of the Brahmanas, Vedas, Vedic virtues, Padma and
Šrī Laksmī. The god of the great refulgence made another set
comprising of Jyesha, Alaksmi, the base of me, excluded from the Vedic
sphere as well as the sin. It was after creating Alaksmi at the outset
that Visnu created Padma. Therefore, Alaksml is Jyestha (the elder).
O excellent Brahmanas, at the outcome of nectar soon after the
terribly potent poison, the inauspicious Jyestha was born. It is so
heard. Thereof, Sri Padma was born, who subsequently became the wife
of lord Visnu.
Alakshmi was later married to Sage Duhsaha. But She couldn't stay with him wherever there was chants of Vedas, chants of names of Hari and Hara and other auspicious things. Duhusha got confused asked sage Markendeya for the reason. Markandeya then described places liked and disliked by her.
This chapter describes in detail all auspicious places, where she can't stay such as places where Vedas and names of Lord are chanted and inauspicious places, where she likes to live like warring families and Adharmic places. (Read that chapter to know them in detail).
Finally Duhusa left her at a tank and went to Patalaloka through an anthill.
दुःसहस्तामुवाचेदं तडागाश्रममंतरे । आस्व त्वमत्र चाहं वै प्रवेक्ष्यामि
रसातलम्॥७७॥ आवयोः स्थानमालोक्य निवासार्थं ततः पुनः । आगमिष्यामि ते
पार्श्वमित्युक्ता तमुवाच सा॥७८॥ किमश्नामि महाभाग को मे दास्यति वै
बलिम्। इत्युक्तस्तां मुनिः प्राह याः स्त्रियस्त्वां यजंति वै॥
Once Duhsaha said to her- “You live at the bank of this tank in the
cottage, because I shall enter Patalaloka. I shall reach for a
suitable place for dwelling of both of us and shall return to you.” At
this, Jyesha said“What shall I do here? Who will feed me here?” At the
words thus spoken by Jyesha, the sage replied to her.
बलिभिः पुष्पधूपैच न तासां च गृह विश। इत्युक्त्वा त्वाविशत्तत्र पातालं
बिलयोगतः॥८०॥ अद्यापि च विनिर्मग्नो मुनिः स जलसंस्तरे।
ग्रामपर्वतबाह्येषु नित्यमास्तेऽशुभा पुनः॥८१॥
“You never enter the houses of the ladies who will adore you offering
incense, flowers and fragrance.” Thus speaking, the sage entered
Patala through an anthill, where the sage lives even today at the
water level. The inauspicious goddess Jyestha always wanders over the
mountains, outside the villages.
Lord Vishnu when spotted Alakshmi alone at tank, He told her to go inauspicious places for food.
ये रुद्रमनघं शर्व शडूरं नीललोहितम्। अंबां हैमवतीं
वापि जनित्रीं जगतामपि॥८५॥ मद्भक्तान्नेिदयंत्यत्र तेषां वित्तं तवैव हि।
येपि चैव महादेवं विर्निद्यैव यजंति माम्॥८६॥ मूढा ह्यभाग्या मन्द्रता
अपि तेषां धनं तव। यस्याज्ञया ह्यहं ब्रह्मा प्रसादाद्वर्तते सदा॥८७॥ ये
यजंति विर्निद्यैव मम विद्वेषकारकाः। मद्भक्ता नैव ते भक्ता इव वर्तीति
दुर्मदाः॥८८॥ तेषां गृहं धनं क्षेत्रमिष्टापूर्तं तवैव हि॥
Lord Visnu said
"Such of the people who do not adore the sinless
Rudra, Sarva, Nīlalohita, Siva and Haimavati - the mother of the
universe, besides my devotees, those who denounce Mahadeva and adore
me, all their riches belong to you. Lord Siva, by adoring Whom, both
myself and Brahma exist, the riches of my foolish devotees belong to
you. Such of the people who are envious of me, denounce me, are not my
devotees and do not Worship me like my devotees, belong to you. Their
riches, the fields, crops and all the noble deeds done by me, belong
to you. The digging of the tanks and Wells also belong to you.”