Siltuximab in Large Granular Lymphocytic Leukemia (LGLL)

Inclusion Criteria: * Provision of signed and dated informed consent form * Stated willingness to comply with all study procedures and availability for the duration of the study. * Male or female, aged \>/= 18. * Meet the diagnosis criteria of LGLL as below: * a. For a patient with treatment-naive LGLL, peripheral blood needs to have CD3+ CD57+ cells \>400/mm³ or CD8+ cells \>650/mm³, and, evidence for clonal T cell receptor gamma or beta gene rearrangement by PCR needs to be detected in either peripheral blood or bone marrow. * b. For patients with an established diagnosis of LGLL who have been previously treated, multicolor flow cytometry should identify a residual CD3+CD57+CD8+ LGLL population in peripheral blood or bone marrow, and, evidence for clonal T cell receptor gamma or beta gene rearrangement by PCR needs to be detected in either peripheral blood or bone marrow. There is no requirement that peripheral blood absolute clonal CD3+ CD57+ or CD8+ LGLL populations need to reach predetermined minimal value for eligibility since immunosuppressive therapy can significantly decrease LGLL cell count without any impact on neutropenia, anemia or thrombocytopenia which are major causes of morbidity and mortality * Has at least one of the indications for treatment: 1. severe neutropenia less than 500/mm³, OR 2. neutropenia associated with recurrent infection, OR 3. symptomatic anemia with Hemoglobin \< 9 g/dL, OR 4. transfusion-dependent anemia with transfusion needs \>= 1 u per month, OR 5. severe thrombocytopenia \<20,000/mm³, OR 6. thrombocytopenia \<50,000/mm³ with bleeding. * Participant can be treatment-naïve or previously treated for LGLL. * Participant currently receiving therapy must have a wash-out period of ≥ 30 days or 5 elimination half-lives, whichever is longer, prior to study drug administration. * Eastern Cooperative Oncology Group (ECOG) performance status 0-2. * Creatinine clearance (CLCr) ≥15 mL/min. * If a participant has chronic liver disease, Child-Pugh score needs to be either A or B. * For females of reproductive potential: use of highly effective contraception for at least 1 month prior to study drug infusion and agreement to use highly effective contraception during study participation and for an additional 3 months after the last dose of study drug. * For males of reproductive potential: use of condoms or other methods to ensure effective contraception with partner during study participation and for an additional 3 months after the last dose of study drug. Men must agree to not donate sperm during the same period. Exclusion Criteria: * Any active infection requiring systemic therapy, including viral infections such as HIV, Hepatitis B, and/or Hepatitis C. * Current use of methotrexate, cyclophosphamide, or cyclosporine for any medical conditions. * Has coexisting myelodysplastic syndrome (MDS). * Elevated LGL due to viral infection. * Pregnancy or lactation. * Known severe allergic reactions to siltuximab. * At increased risk for Gastrointestinal (GI) perforation, in the opinion of the study investigator. * Received live vaccine 30 days prior to study drug administration or Intend to receive live vaccine during treatment period and within 3 months after last dose of study drug. * Rheumatological conditions such as rheumatoid arthritis (RA) are not exclusion criteria for the study. * Coexisting hematological conditions such as autoimmune hematological anemia (AIHA) or immune thrombocytopenia (ITP) are not automatic exclusion criteria but will be at discretion of study investigator. * Previous or concurrent malignancies not considered cured, except inactive non-melanoma skin cancer, in situ carcinoma of the cervix, early-stage prostate cancer, or other cancer deemed clinically insignificant by the Investigator. * Has a history or current evidence of any condition, therapy, or laboratory abnormality that might confound the results of the trial, interfere with the participant's participation for the full duration of the trial, or is not in the best interest of the participant to participate, in the opinion of the study Investigator.